Find the values of m and b that make the following function differentiable.
the piecewise function f of x equals x squared when x is less than or equal to three or mx plus b when x is greater than three
Answers
For [tex]f[/tex] to be differentiable, it must be continuous, so we need to have
[tex]\displaystyle\lim_{x\to3^-}f(x)=\lim_{x\to3^+}f(x)=f(3)[/tex]
By its definition, [tex]f(3)=3^2=9[/tex]. The one-sided limits are
[tex]\displaystyle\lim_{x\to3^-}f(x)=\lim_{x\to3}x^2=9[/tex]
[tex]\displaystyle\lim_{x\to3^+}f(x)=\lim_{x\to3}mx+b=3m+b[/tex]
so we require [tex]3m+b=9[/tex].
In order for [tex]f[/tex] to be differentiable at [tex]x=3[/tex], we also need to have [tex]f'(3)[/tex] exist, which requires that [tex]f'[/tex] also be continuous at [tex]x=3[/tex]. First, compute the derivatives of all pieces of [tex]f[/tex]:
[tex]f'(x)=\begin{cases}2x&\text{for }x<3\\?&\text{for }x=3\\m&\text{for }x>3\end{cases}[/tex]
[tex]f'[/tex] is continuous at [tex]x=3[/tex] if
[tex]\displaystyle\lim_{x\to3^-}f'(x)=\lim_{x\to3^+}f'(x)=f'(3)[/tex]
The one-side limits are
[tex]\displaystyle\lim_{x\to3^-}f'(x)=\lim_{x\to3}2x=6[/tex]
[tex]\displaystyle\lim_{x\to3^+}f'(x)=\lim_{x\to3}m=m[/tex]
so we need to have [tex]m=6[/tex], and moreover [tex]f[/tex] will be differentiable if we set [tex]f'(3)=6[/tex].
So with [tex]m=6[/tex], we must have [tex]3m+b=9\implies b=-9[/tex].
For the piecewise function f(x) = x^2 when x <= 3 and mx + b when x > 3, values m = 6 and b = -9 ensure differentiability at x = 3.
To make the piecewise function f(x) differentiable at x = 3, we need the two pieces, x^2 and mx + b, to smoothly connect at x = 3. This requires the values of m and b to ensure continuity of both function values and derivatives.
First, evaluate both pieces at x = 3:
x^2 when x <= 3 and mx + b when x > 3
For continuity, set these expressions equal to each other:
3^2 = m * 3 + b
This yields 9 = 3m + b. To ensure differentiability, the derivatives of both pieces must also match at x = 3:
f'(x) = 2x when x <= 3 and f'(x) = m when x > 3
For continuity of derivatives, set the derivatives equal to each other at x = 3:
2 * 3 = m
This gives m = 6. Substituting m = 6 into the continuity equation 9 = 3m + b gives b = -9.
Therefore, the values m = 6 and b = -9 make the piecewise function f(x) differentiable at x = 3.
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Related Questions
On the first play of the game, the Wildcats gained 8 yd. What integer represents a gain of 8 yd? Enter your answer in the box.
Answers
Answer:
+8
Step-by-step explanation:
They gained 8 yards so it is +8.
Answer: + 8
Step-by-step explanation: A gain is considered a good thing. It is represented with a positive integer. Here, the gain is represented by +8, or simply 8.
The length of a rectangular is 5 centimeters less then it width. The perimeter of the rectangle is 26
Answers
Hi there! :)
Answer:
Length = 4 cm
Width = 9 cm
Step-by-step explanation:
Here's the information we know:
- Perimeter = 26 cm
- Width = x (I just replaced the width by "x" because it's what we are looking for)
- Length = x - 5
Now before we can do anything, you need to know what the formula for the perimeter of a rectangular is: P = (2 × Length) + (2 × Width)
Replace all the information you know in the formula and solve the equation by isolating "x":
P = (2 × Length) + (2 × Width)
26 = ( 2 × (x - 5)) + (2 × x)
Multiply "2" and "x" together → 2 × x = 2x
26 = ( 2(x - 5)) + 2x
Distribute the "2" to the entire parentheses → 2 × x = 2x & 2 × -5 = -10
26 = 2x - 10 + 2x
Add similar terms together → 2x + 2x = 4x
26 = 4x - 10
Add 10 to each side of the equation → 26 + 10 = 36
36 = 4x
Divide each side of the equation by 4 → 36 ÷ 4 = 9
9 = x ⇒ Measure of the width
Since the lenght is 5 centimeter less than the measure of the width, in order to find the measure of the length you'll need to subtract "5" from the measure of the width, which is "9":
9 - 5 = Length rectangle
9 - 5 = 4 ⇒ Length rectangle
There you go! I really hope this helped, if there's anything just let me know! :)
The nicotine content in cigarettes of a certain brand is normally distributed with mean
(in milligrams) m and standard deviation s = 0.1. The brand advertises that the mean
nicotine content of their cigarettes is 1.5, but measurements on a random sample of 400
cigarettes of this brand gave a mean of x = 1.52. Is this evidence that the mean nicotine
content is actually higher than advertised? To answer this, test the hypotheses
H0: m = 1.5, Ha: m > 1.5
at significance level a = 0.01. You conclude
A) that H0 should be rejected.
B) that H0 should not be rejected.
C) that Ha should be rejected.
D) that there is a 5% chance that the null hypothesis is true.
Answers
Through hypothesis testing, we cannot reject the null hypothesis at the 1% level of significance. Thus, there is no strong evidence to suggest that the mean nicotine content is greater than what is advertised.
Explanation:The subject here is statistics, specifically hypothesis testing. For this problem, we are asked to test the null hypothesis H0: m = 1.5 against the alternative hypothesis Ha: m > 1.5 at 1% level of significance.
Using the given data, we calculate the test statistic (z) using the formula: z = (x - µ) / (s/√n), where x is the sample mean, µ is the population mean, s is the standard deviation, and n is the sample size. Substituting the given values, we get z = (1.52 - 1.5) / (0.1/√400) = 2.
The z-score at 0.01 level of significance is -2.33 on the left and 2.33 on the right. As our test statistic (2) falls within this range, we fail to reject the null hypothesis (H0: m = 1.5). This suggests that there's no strong evidence to support that the mean nicotine content is greater than advertised.
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What is the probability for flipping a tail with a coin and rolling a four with a die
Answers
A recursive rule for a geometric sequence is a1=2; an= 4/9 an-1
What is the explicit rule for this sequence?
Answers
A recursive rule for a geometric sequence:
[tex]a_1\\\\a_n=r\cdot a_{n-1}[/tex]
---------------------------------------------------
[tex]a_1=2;\ a_n=\dfrac{4}{9}a_{n-1}\to r=\dfrac{4}{9}[/tex]
---------------------------------------------------
The exciplit rule:
[tex]a_n=a_1r^{n-1}[/tex]
Substitute:
[tex]a_n=2\left(\dfrac{4}{9}\right)^{n-1}=2\cdot\left(\dfrac{4}{9}\right)^n\cdot\left(\dfrac{4}{9}\right)^{-1}=2\cdot\left(\dfrac{4}{9}\right)^n\cdot\dfrac{9}{4}\\\\\boxed{a_n=\dfrac{9}{2}\cdot\left(\dfrac{4}{9}\right)^n}[/tex]
The following is an illustration of a regular polygon:
Answers
Answer:
False
Step-by-step explanation:
A regular sided polygon means all the sides have the same length and all angles the same size. This polygon has 2 sides the same length, not 3 sides the same length, so it cannot be regular.
Thanks in advance for your help, the question is;
Half the first, plus twice the second of two consecutive integers equals 22. What are the two integerrs?
Answers
Answer:
8, 9
Step-by-step explanation:
1/2x + 2(x + 1) = 22
1/2x + 2x + 2 = 22
1/2x + 2x = 20
5/2x = 20
5x = 40
x = 8
8, 9
4 + 18 = 22
Answer:
8 and 9
Step-by-step explanation:
hope this helps!
What is the scale factor of △UVW to △ XYZ?
Answers
Answer:
B. 1/7
Step-by-step explanation:
The triangles are similar, so the ratios of corresponding sides are the same.
... UV/XY = 3/21 = 1/7 . . . . . matches B
_____
When we talk about the ratio "A to B", we usually mean A:B or A/B. There's the possibility that the question could be interpreted to mean, "what factor do I multiply by to get from A to B?" In that case, the result here would be 7, because XY is 7 times the length of UV.
What is the amplitude of the function shown in the graph? Enter your answer in the box.
Answers
Answer: A = 2.5
Step-by-step explanation:
The highest point is 5.5. The lowest point is 0.5.
[tex]Amplitude (A) = \dfrac{high-low}{2}[/tex]
[tex]A = \dfrac{5.5-0.5}{2}[/tex]
[tex]= \dfrac{5}{2}[/tex]
[tex]= 2.5[/tex]
Bridget drives at a speed of 60 miles per hour. how long will it take to drive 270 miles
Answers
Divide the total miles by her speed:
270 miles / 60 miles per hour = 4.5 hours.
Given that SU≅UT and SV>VT, identify a two-column proof that proves m∠SUV>m∠TUV.
Answers
Answer: Choice B
We start with SU = UT being given, and so is SV > VT.
The second line is UV = UV by the reflexive property to help us use the converse of the hinge theorem next.
The hinge theorem basically says that the more open an angle is, the larger the opposite side will be (the side opposite from the angle that varies). Imagine a door opening and closing. The further open it gets, the longer the rope will be that connects the door handle to the frame. Going in reverse, the converse says that if the opposite side of the angle gets bigger, then the angle itself gets bigger as well. The hinge theorem is handy to help compare two sides even if we don't know their exact lengths (we can figure out which side is larger than the other)
The conditions given that SU is congruent to UT and SV is greater than VT create a scenario where we can prove that the measure of angle SUV is greater than the measure of angle TUV.
Explanation:Given that SU≅UT means that segment SU is congruent to segment UT, and SV>VT meaning that segment SV is greater than segment VT, this sets up two important conditions for the triangle SUV. First, as SU is congruent to UT, this implies that ∠SUV is congruent to ∠TUV due to the Base Angles Theorem (If two sides of a triangle are congruent (equal), then angles opposite these sides are congruent). However, SV>VT creates an inequality which contradicts this, pointing towards the fact that ∠SUV is greater than ∠TUV.
Proof:
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Each CD in a store costs $10, and each book costs $6. If you want to spend exactly $32, write an equation in standard form modeling this situation. Let c represent the number of CDs you buy, and b represent the number of books you buy.
Answers
c2+b2=32
Hope this helps :) good luck!
Final answer:
To model the situation with an equation, use the quantities of CDs and books as variables c and b, and their prices to create the equation 10c + 6b = 32, which represents spending exactly $32.
Explanation:
To create an equation modeling the situation where you want to spend exactly $32 on CDs and books, we need to use the given prices for each item. Let c represent the number of CDs you buy at $10 each, and let b represent the number of books you buy at $6 each.
The total amount spent will be the sum of the amount spent on CDs and the amount spent on books. Using the price of each item, we can write the following equation:
10c + 6b = 32
This equation can now be written in standard form, which is Ax + By = C, where A, B, and C are integers, and A is non-negative. Therefore, our equation in standard form is:
10c + 6b = 32
Please answer quickly ill give Brainliest.
Answers
The slope-intercept form:
[tex]y=mx+b[/tex]
m - slope
b - y-intercept → (0, b)
Read two points from graph
y-intercept: (0, -3) → b= -3
x-intercept: (-2, 0).
The formula of a slope:
[tex]m=\dfrac{y_2-y_1}{x_2-x_1}[/tex]
Substitute:
[tex]m=\dfrac{0-(-3)}{-2-0}=\dfrac{3}{-2}=-\dfrac{3}{2}[/tex]
Therefore we have the equation of a line in slope-intercept form:
[tex]y=-\dfrac{3}{2}x-3[/tex]
Convert to the standard form [tex]Ax+By=C[/tex]
[tex]y=-\dfrac{3}{2}x-3[/tex] multiply both sides by 2
[tex]2y=-3x-6[/tex] add 3x to both sides
[tex]3x+2y=-6[/tex]
Answer: 3x + 2y = -6Answer:
y=mx+b
Step-by-step explanation:
7.6g of sugar is needed for every cake made. How much sugar is needed for 5 cakes
Answers
Answer:
38 grams
Step-by-step explanation:
7.6*5=38
Jim reads 3 newspaper articles every day, and Jessica reads 5 times more than Jim does. How many newspaper articles does Jessica read every day?
Answers
Answer:
Jessica reads 15 newspaper articles
Step-by-step explanation:
Jim reads 3
Jessica read 5 times MORE than Jim
5 x 3= 15
So Jessica reads 15 articles
Jim reads (3)
Jessica reads (5x more)
(3x5 = 15)
A car salesman had $65100 in total monthly sales last month. He made $1953 in commission from those sales. What is the salesmans commission as a percent of his total monthly sales?
Answers
Answer:
3%
Step-by-step explanation:
In order to find out the percent in commissions a car salesman received, you need to divide $1953 by $65100. The result is .03, which translates to 3%.
Hope that helps!
What is the rate of change from x = 0 to x = pi over 2?
trig graph with points at 0, negative 4 and pi over 2, 0 and pi, 4 and 3 pi over 2, 0 and 2 pi, negative 4
Answers
Answer:
8/pi
Step-by-step explanation:
To find the rate of change from 0 to pi/2 we need the values at 0 and pi/2
f(0) = -4
f(pi/2) = 0
rate of change = change in f(x)/ change in x
= (f(x2) -f(x1))/( x2-x1)
= (0--4) / (pi/2 - 0)
= (0+4) / (pi/2)
= 4 / (pi/2)
= 8/pi
What is the 31st term of the arithmetic sequence for -15 -11 -7 -3
Answers
Answer:
its -11
Step-by-step explanation:
Oliver runs for 9 more minutes than Bobby.
The equation v = 9 + b, where v represents the number of minutes Oliver runs, and b represents the number of minutes Bobby runs, shows this relationship.
If Oliver runs 33 minutes, how many minutes does Bobby run?
A. 15
B. 42
C. 24
D. 51
Answers
If Oliver runs 9 minutes more, and we already know how long he ran, then we need to do the opposite of the equation, which is to subtract instead of add. 33-9=24
What type of triangle has its circumcenter, its incenter, its centroid, and its orthocenter all at the same point?
Answers
The only triangle with the usual triangle centers coincident is an equilateral triangle.
The centroid is the meet of the medians, which are the segments from a vertex to the midpoint of its opposite side. The centroid is the balance point of the triangle. It always divides the medians into a 2:1 ratio, the long piece connected to the vertex.
The circumcenter is the center of the circumcircle, which is the circle through the vertices. That's only going to be the centroid when that 2 part of the 2:1 ratio is the same for each median. That's only going to happen when the sides are the same length so the centroid makes congruent triangles.
The incenter is the center of the inscribed circle, which is the meet of the angle bisectors of each vertex. Again it wont be the centroid unless the sides are equal.
The orthocenter is the meet of the altitudes and it's the same story.
An equilateral triangle is the only type of triangle where its circumcenter, incenter, centroid, and orthocenter coincide at the same point. This unique property emphasizes the symmetry of the equilateral triangle.
Explanation:The type of triangle where its circumcenter, incenter, centroid, and orthocenter are all at the same point is an equilateral triangle. In an equilateral triangle, all sides are equal in length, and all angles are equal to 60 degrees.
An equilateral triangle has the unique property that its circumcenter (intersection of perpendicular bisectors of sides), its incenter (intersection of angle bisectors), its centroid (intersection of medians), and its orthocenter (intersection of altitudes) all coincide at the same point, which is the geometric center and also the triangle's center of mass.
This property is exclusive to equilateral triangles and emphasizes their symmetry. This also applies in three-dimensional forms such as in an icosahedron where each face is an equilateral triangle, demonstrating further the symmetrical properties of these shapes.
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Marry had 22 3/4 grams of salt. She used 7 1/4 grams to make spinach omelets and 4 7/8 grams to make mushroom omelets. About how much salt does Marry have left? Choose the best estimate.
Answers
Answer:
10 5/8
Step-by-step explanation:
We need to subtract (7 1/4 + 4 7/8) from 22 3/4 grams of salt.
We compute the whole numbers and fractions separately:-
7 + 4 = 11
1`/4 + 7/8 = 2/8 + 7/8 = 9/8 = 1 1/8.
So 7 1/4 + 4 7/8 = 12 1/8
Therefore:-
22 3/4 - 12 1/8
22 - 12 = 10
3/4 - 1/8 = 5/8
= 10 + 5/8 = 10 5/8 (answer)
If there are 25,682 people living in Jarvieville in 2015, and the population increases 20% each year...
a) write the equation to use to predict future years
b) how many people will there be in this town in the year 2018?
Answers
Answer:
[tex]f(x)=25,682(1.2)^{t}[/tex]
44378.496=44,379
Step-by-step explanation:
We can write the equation using a growth equation [tex]f(x)=P(1+r)^{t}[/tex] where:
P is the starting valuer is the percent or rate as a decimalt is yearsWe will substitute into the formula P=25,682, r=0.2 and t =3 for 3 years since 2018 is 3 years after 2015.
[tex]f(x)=25,682(1+0.2)^{3}[/tex]
[tex]f(x)=25,682(1.2)^{3}[/tex]
[tex]f(x)=44378.496[/tex]
The graph of an absolute value function opens down and has a vertex of (-3,0).
The domain of the function is ?
The range of the function is ?
Answers
The domain of the function is:
All real numbers i.e. (-∞,∞)
The range of the function is:
(-∞,0]
Step-by-step explanation:It is given that:
The graph of an absolute value function opens down and has a vertex of (-3,0).
This means that the graph of the function increases continuously in the interval (-∞,-3] and takes maximum value 0 when x= -3 and then decreases continuously in the interval (-3,∞).
Domain of a function--
It is the set of all the x-values for which a function is defined i.e. it is the set of all the values of the independent variable for which the function is defined.
Range of a function--
It is the set of all the values attained by a function.
We know that a absolute value function is defined all over the real line i.e. the domain of the function is the set of all the real values i.e. (-∞,∞).Also, the function takes all the value between -∞ and 0 and 0 is included.Hence, the range of the function is: (-∞,0].
From the table below, determine whether the data shows an exponential function. Explain why or why not.
Answers
Answer:
B
Step-by-step explanation:
A local bake shop sells cookies and cakes.On monday , sean bought 3 cookies and 5 cakes and spent $78.75 on tuesday , sean bought 8 cookies and 2 cakes and spent $40 . Find the price of one cookie and one cake .
Answers
Answer:
The price of one cookie is $1.25 and the price of one cake is $15.
Step-by-step explanation:
Let $x be the price of one cookie and $y be the price of one cake.
1. On Monday, Sean bought 3 cookies and 5 cakes, then he spent $(3x+5y) and this is $78.75. Thus,
[tex]3x+5y=78.75.[/tex]
2. On Tuesday, Sean bought 8 cookies and 2 cakes, then he spent $(8x+2y) and this is $40. Thus,
[tex]8x+2y=40.[/tex]
3. Solve the system of two equations:
[tex]\left\{\begin{array}{l}3x+5y=78.75\\8x+2y=40\end{array}\right.[/tex]
Multiply the first equation by 2 and the second equation by 5 and subtract them:
[tex]2(3x+5y)-5(8x+2y)=2\cdot 78.75-5\cdot 40,\\ \\6x+10y-40x-10y=157.5-200,\\ \\-34x=-42.5,\\ \\x=\$1.25.[/tex]
Then
[tex]8\cdot 1.25+2y=40,\\ \\2y=40-10,\\ \\y=\$15.[/tex]
a shoes has a length of 11 inches. Which unit conversion fraction should you use to find the length in centimeters?
A. 2.54 cm/1 in
B. 1 cm/2.54 cm
C. 1 in / 2.54 cm
D. 1 in / 1 cm
Answers
Answer:
2.54 cm/1 inch
Step-by-step explanation:
We have inches. That is in the numerator. We need to convert to cm. So our conversion factor should have cm in the numerator and inches in the denominator.
1 in = 2.54 cm
Divide both sides by 1 inc
A conversion factor is equal to 1
1 = 2.54 cm/1 inch
The correct unit conversion fraction to convert inches to centimeters is 2.54 cm/1 in. Apply this factor to the length in inches (11 inches) to obtain the length in centimeters (27.94 cm).
Explanation:To convert a length in inches to centimeters, we can use a unit conversion fraction. In this case, because there are 2.54 centimeters in one inch, the correct unit conversion fraction to use is A. 2.54 cm/1 in. This means that we multiply the length in inches by the conversion factor to get the length in centimeters. Here is the step-by-step process for your problem:
Convert the given length to centimeters using the correct conversion fraction : Length in centimeters = Length in inches * Conversion factor Length in centimeters = 11 inches * 2.54 cm/in Length in centimeters = 27.94 cmLearn more about Unit Conversion here:https://brainly.com/question/19420601
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Identify the values of the variables. Give your answers in simplest radical form. PLEASE HELP!!
Answers
Answer:
2 sqrt(3) = n
k = 4 sqrt(3)
Step-by-step explanation:
We know that cos 30 = adjacent/ hypotenuse
cos 30 = 6 /k
Multiply each side by k
k cos 30 = 6k * k
k cos 30 = 6
Divide each side by cos 30
k cos 30/ cos 30 = 6/ cos 30
k = 6 / cos 30
k = 6/ (sqrt(3)/2)
k =12 / sqrt(3)
We cannot leave the sqrt in the denominator
k = 12 / sqrt(3) * sqrt(3)/sqrt(3)
k = 12 sqrt(3)/3
k = 4 sqrt(3)
We know tan 30 = opposite/adjacent
tan 30 = n/6
Multiply each side by 6
6 tan 30 = n
6 * (sqrt(3)/3) = n
6/3 * sqrt(3) = n
2 sqrt(3) = n
The correct option is d. The values are 2 sqrt(3) = n, k = 4 sqrt(3)
We know that cos 30 = adjacent/ hypotenuse
Multiply each side by k
k cos 30 = 6k * k
k cos 30 = 6
Divide each side by cos 30
k cos 30/ cos 30 = 6/ cos 30
k = 6 / cos 30
k = 6/ (sqrt(3)/2)
k =12 / sqrt(3)
We cannot leave the sqrt in the denominator
k = 12 / sqrt(3) * sqrt(3)/sqrt(3)
k = 12 sqrt(3)/3
k = 4 sqrt(3)
We know tan 30 = opposite/adjacent
tan 30 = n/6
Multiply each side by 6
6 tan 30 = n
6 * (sqrt(3)/3) = n
6/3 * sqrt(3) = n
2 sqrt(3) = n
The price of gold is often reported per ounce. At the end of 2005, this price was $513. At the end of 2015, it was $1060. By what percentage did the price per ounce of gold increase?
Answers
Answer:
106.63%.
Step-by-step explanation:
We have been given that at the end of 2005, this price was $513. At the end of 2015, it was $1060.
To find the percentage of the price per ounce of gold increase we will use % increase formula.
[tex]\text{Percent Increase}=\frac{\text{Final value-Initial value}}{\text{Initial value}}\times 100[/tex]
Let us substitute our given values in above formula.
[tex]\text{Percent Increase}=\frac{1060-513}{513}\times 100[/tex]
[tex]\text{Percent Increase}=\frac{547}{513}\times 100[/tex]
[tex]\text{Percent Increase}=1.0662768031189084\times 100[/tex]
[tex]\text{Percent Increase}=106.62768031189084\approx 106.63[/tex]
Therefore, the price of per ounce of gold increased by 106.63%.
Answer:107
Step-by-step explanation:
So if it is incorrect for the other answer this is the answer, so do 1060-513=547, so it increased by that much, then divided 547/513= 1.06627680312 But that rounds up to 7 and since we know that the original was 100% of its amount we would add 100+7 which would be 107%
A recipe for tropical punch calls for 2 cups of pineapple juice and 3 cups of orange juice. Jo creates a drink by mixing 3 cups of pineapple juice with 4 cups of orange juice. How does Jo's drink compare to the tropical punch recipe?
Use the Pic if Needed :)
Answers
Answer:
Jo's drink is bigger than the tropical punch, both have more orange juice than pineapple.
Step-by-step explanation:
Tropical punch is 2 cups of pineapple juice and 3 cups of orange juice
We express those numbers is fraction , to get the denominator we add the 2 parts, 2 cups + 3 cups= 5
Then,
pineapple juice=2/5
orange juice=3/5
Drink is 3 cups of pineapple juice with 4 cups of orange juice
Denominator is 3 + 4 = 7
pineapple juice=3/7
orange juice=4/7
Jo's drink is bigger than the tropical punch, both have more orange juice than pineapple.
Which equation has exactly two real and two non real solutions?
Answers
Answer:
The first option [tex]x^4-21x^2-100=0[/tex].
Step-by-step explanation:
To have exactly 2 real and two non real solutions, the degree of the polynomial must be a degree 4. Degree is the highest exponent value in the polynomial and is also the number of solutions to the polynomial. This polynomial ha 2 real+2 non real= 4 solutions and must be [tex]x^4[/tex]. This eliminates the bottom two solutions.
In order to have two real and two non real solutions, the polynomial must factor. If it factors all the way like
[tex]x^4-100x^2=0\\x^2(x^2-100)=0\\x^2(x-10)(x+10)=0\\\\x^2=0\\x-10=0\\x+10=0[/tex]
This means x=0, 10, -10 are real solutions to the polynomial. It has no non real solutions. This eliminates this answer choice.
Only answer choice 1 meets the requirement.
Answer:
A. x^4 - 21x^2 - 100 = 0