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Precalculus : Derivatives

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Example Questions

Example Question #1 : Find The Critical Numbers Of A Function

Find the critical numbers of the following function:

$f(x)=\frac{1}{3}x^3-2x^2-12x$

Possible Answers:

x=-6,x=2

$x=\frac{1}{3},x=2$

x=-2,x=6

x=5,x=7

x=-7,x=-5

Correct answer:

x=-2,x=6

Explanation:

The critical numbers of a function are those at which its first derivative is equal to 0. These points tell where the slope of the function is 0, which lets us know where the minimums and maximums of the function are. First we find the derivative of the function, then we set it equal to 0 and solve for the critical numbers:

$f(x)=\frac{1}{3}x^3-2x^2-12x$

f'(x)=x^2-4x-12=(x-6)(x+2)=0

x=-2,x=6

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Example Question #7 : Find The Critical Numbers Of A Function

Find the critical numbers of the following function:

$f(x)=\frac{2}{3}x^3-5x^2+12x$

Possible Answers:

x=-3,x=-2

x=2,x=3

x=5,x=12

x=-12,x=5

$x=-\frac{2}{3},x=4$

Correct answer:

x=2,x=3

Explanation:

$f(x)=\frac{2}{3}x^3-5x^2+12x$

f'(x)=2x^2-10x+12=2(x^2-5x+6)=2(x-2)(x-3)=0

x=2,x=3

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Example Question #1 : Find The Critical Numbers Of A Function

Find the critical numbers of the following function:

$f(x)=x^3+\frac{3}{2}x^2-90x$

Possible Answers:

x=-3,x=4

x=-5,x=6

x=-4,x=3

$x=-\frac{3}{4},x=1$

x=-6,x=5

Correct answer:

x=-6,x=5

Explanation:

$f(x)=x^3+\frac{3}{2}x^2-90x$

f'(x)=3x^2+3x-90=3(x^2+x-30)=3(x+6)(x-5)=0

x=-6,x=5

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Example Question #1 : Find The Critical Numbers Of A Function

Find the critical value(s) of the function $f(x)=3x^{3}$ .

Possible Answers:

x=0,3

x=0

x=0,-3

Answer not listed

x=3,-3

Correct answer:

x=0

Explanation:

The critical values of a function f(x) are values for which the derivative f'(x)=0 . In this case:

$f(x)=3x^{3}$

$f'(x)=9x^{2}$

Setting f'(x)=0 :

$9x^{2}=0$

$x^{2}=0$

x=0

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Example Question #1 : Find The Critical Numbers Of A Function

Find the critical value(s) of the function $f(x)=x^{2}-2x+1$ .

Possible Answers:

Correct answer not listed

x=1

x=1,0

x=1,-1

x=-1

Correct answer:

x=1

Explanation:

The critical values of a function f(x) are values for which the derivative f'(x)=0 . In this case:

$f(x)=x^{2}-2x+1$

f'(x)=2x-2

Setting f'(x)=0 :

2x-2=0

2x=2

x=1

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Example Question #101 : Derivatives

Find the critical value(s) of a function $f(x)=(x+1)^{3}$ .

Possible Answers:

x=-1

x=0,1

x=1,-1

x=0,-1

x=1

Correct answer:

x=-1

Explanation:

The critical values of a function f(x) are values for which the derivative f'(x)=0 . In this case:

$f(x)=(x+1)^{3}$

$f'(x)=3(x+1)^{2}$

Setting f'(x)=0 :

$3(x+1)^{2}=0$

$(x+1)^{2}=0$

x+1=0

x=-1

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Example Question #101 : Derivatives

Find the critical numbers of the following function:

f(x)=x^3-5x^2+4x+9

Possible Answers:

$x=3-\sqrt{6},x=3+\sqrt{6}$

$x=\frac{1}{3}(5-\sqrt{13}),x=\frac{1}{3}(5+\sqrt{13})$

x=-3, x=4

$x=\frac{1}{2}(7-3\sqrt{5}),x=\frac{1}{2}(7+3\sqrt{5})$

$x=10-\sqrt{11},x=10+\sqrt{11}$

Correct answer:

$x=\frac{1}{3}(5-\sqrt{13}),x=\frac{1}{3}(5+\sqrt{13})$

Explanation:

The critical numbers of a function are the points at which its slope is zero, which tells where the function has a minimum or maximum. The slope of a function is described by its derivative, so we'll take the derivative of the function and set it equal to 0 to solve for the x values of the critical numbers:

f(x)=x^3-5x^2+4x+9

f'(x)=3x^2-10x+4

It appears the derivative cannot be factored to solve for x, so we'll have to use the quadratic formula to find the critical numbers:

3x^2-10x+4=0

$\frac{10\pm \sqrt{(-10)^2-4(3)(4)}}{2(3)}=\frac{5}{3}\pm\frac{1}{3}\sqrt{13}=\frac{1}{3}(5\pm \sqrt{13})$

So the critical numbers occur at the following two values of x:

$x=\frac{1}{3}(5-\sqrt{13}),x=\frac{1}{3}(5+\sqrt{13})$

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Example Question #771 : Pre Calculus

Given the following function, find the critical numbers: y= 3x^2-2x-50

Possible Answers:

$-\frac{2}{3}$

$-\frac{1}{3}$

$\frac{1}{3}$

Correct answer:

$\frac{1}{3}$

Explanation:

Critical numbers are where the slope of the function is equal to zero or undefined.

Find the derivative and set the derivative function to zero.

y= 3x^2-2x-50

y'= 6x-2 = 0

6x=2

$x=\frac{1}{3}$

There is only one critical value at $x=\frac{1}{3}$ .

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Example Question #103 : Derivatives

Determine the critical numbers of the function

$\small f(x)=\frac{x^2}{2}-x$

Possible Answers:

$\small DNE$

$\small 1,0$

$\small 1$

$\small 2$

Correct answer:

$\small 1$

Explanation:

The critical numbers are the $\small c$ values for which either

$\small \small f'(c)=0$ or $\small \small f'(c)$ is undefined.

In order to find the first derivative we use the power rule which states

$\small \frac{\mathrm{d} }{\mathrm{d} x} x^n = nx^{n-1}$

Applying this rule term by term we get

$\small f'(x)=\frac{\mathrm{d} }{\mathrm{d} x}[\frac{x^2}{2}-x]$

$\small =2\cdot\frac{x^{2-1}}{2}-1\cdot x^{1-1}$

$\small =x-1$

The first derivative is defined for all values of x. Setting the first derivative to zero yields

$\small f'(x)=x-1=0$

$\small x=1$

As such, the critical number is

$\small 1$

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Example Question #15 : Find The Critical Numbers Of A Function

Find the critical values of the following function:

f(x)=x^2-4

Possible Answers:

none

Correct answer:

Explanation:

To solve, simply find the first derivative and find when it is equal to 0. To find the first derivative, we must use the power rule as outlind below.

Power rule: $(x^n)'=nx^{n-1}$

Thus,

f(x)=x^2-4

$f'(x)=2x^{2-1}-4x^{1-1}=2x^1-4x^0=2x-4$

Now, we must set out function equal to 0 and solve for x. Thus,

2x=0

Dividing both sides by 2, we get:

x=0

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Precalculus : Derivatives

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Find The Critical Numbers Of A Function

Example Question #7 : Find The Critical Numbers Of A Function

Example Question #1 : Find The Critical Numbers Of A Function

Example Question #1 : Find The Critical Numbers Of A Function

Example Question #1 : Find The Critical Numbers Of A Function

Example Question #101 : Derivatives

Example Question #101 : Derivatives

Example Question #771 : Pre Calculus

Example Question #103 : Derivatives

Example Question #15 : Find The Critical Numbers Of A Function

All Precalculus Resources

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