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Precalculus : Find the Critical Numbers of a Function

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Example Questions

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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Example Question #111 : Derivatives

Find the critical values of the following function.

f(x)=x^2-6x+3

Possible Answers:

Correct answer:

Explanation:

To solve, simply differentiate using the power rule, as outlined below.

Power rule states,

$(x^n)'=nx^{n-1}$ .

Thus given,

f(x)=x^2-6x+3

our first derivative is:

$f'(x)=2x^{2-1}-6x^{1-1}=2x-6$

Then plug in 0 for f(x) to find when our function is equal to 0.

Thus,

0=2x-6

2x=6

x=3

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Precalculus : Find the Critical Numbers of a Function

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

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

Example Question #111 : Derivatives

All Precalculus Resources

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