Math  /  Calculus

QuestionLet f(x)=1x24f(x)=\frac{1}{x^{2}-4} Where does ff have critical points? Choose all answers that apply:
A x=2x=-2 B x=0x=0 c. x=2x=2
D ff has no critical points.

Studdy Solution

STEP 1

What is this asking? Find all the *x*-values where the slope of 1x24\frac{1}{x^2 - 4} is zero, or where the function is not differentiable. Watch out! Remember, critical points can occur where the derivative is zero *or* undefined!
Don't forget to check for both!

STEP 2

1. Define the function
2. Calculate the derivative
3. Find where the derivative is zero
4. Find where the derivative is undefined
5. Identify critical points

STEP 3

We're given the function f(x)=1x24f(x) = \frac{1}{x^2 - 4}.
This is our starting point.
Super important to understand what this function represents!

STEP 4

To find the critical points, we need the derivative.
Let's use the **power rule** after rewriting f(x)f(x) as (x24)1(x^2 - 4)^{-1}.
Remember, the power rule says that if g(x)=xng(x) = x^n, then g(x)=nxn1g'(x) = n \cdot x^{n-1}.
We also need the **chain rule**, which says that the derivative of a composition of functions like f(g(x))f(g(x)) is f(g(x))g(x)f'(g(x)) \cdot g'(x).

STEP 5

Applying the power rule and chain rule, we get: f(x)=1(x24)2(2x)f'(x) = -1 \cdot (x^2 - 4)^{-2} \cdot (2x) f(x)=2x(x24)2f'(x) = \frac{-2x}{(x^2 - 4)^2}This tells us the **instantaneous rate of change** of our function at any given *x*-value!

STEP 6

A fraction is zero only when its **numerator** is zero.
So, we need to solve 2x=0-2x = 0.

STEP 7

Dividing both sides by 2-2 gives us x=0x = 0.
So, the derivative is zero at x=0x = 0.

STEP 8

The derivative is undefined when the **denominator** is zero.
We need to solve (x24)2=0(x^2 - 4)^2 = 0.

STEP 9

Taking the square root of both sides gives x24=0x^2 - 4 = 0.

STEP 10

Adding 44 to both sides gives x2=4x^2 = 4.

STEP 11

Taking the square root of both sides gives x=±2x = \pm 2, so x=2x = 2 and x=2x = -2.

STEP 12

We found that the derivative is zero at x=0x = 0 and undefined at x=2x = 2 and x=2x = -2.
These are our **critical points**!

STEP 13

The critical points are x=2x = -2, x=0x = 0, and x=2x = 2.
So the correct answers are A, B, and C.

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