Find all critical points for the function $$ f(x)=-8 x^{2}+64 x-2 $$. Separate multiple answers with a comma.

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UpStudy AI · Accepted Answer

To find the critical points of the function $$ f(x) = -8x^2 + 64x - 2 $$, we need to follow these steps:

1. **Find the derivative** of the function $$ f(x) $$.
2. **Set the derivative equal to zero** to find the critical points.
3. **Solve for $$ x $$**.

Let's start by calculating the derivative of the function. The derivative $$ f'(x) $$ is given by:

$$
f'(x) = \frac{d}{dx}(-8x^2 + 64x - 2)
$$

Now, I will calculate the derivative.
Simplify the expression by following steps:
- step0: Solution:
  $$-8\times 2x+64$$
- step1: Multiply the terms:
  $$-16x+64$$
The derivative of the function is:

$$
f'(x) = -16x + 64
$$

Next, we set the derivative equal to zero to find the critical points:

$$
-16x + 64 = 0
$$

Now, let's solve for $$ x $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$-16x+64=0$$
- step1: Move the constant to the right side:
  $$-16x=0-64$$
- step2: Remove 0:
  $$-16x=-64$$
- step3: Change the signs:
  $$16x=64$$
- step4: Divide both sides:
  $$\frac{16x}{16}=\frac{64}{16}$$
- step5: Divide the numbers:
  $$x=4$$
The critical point is:

$$
x = 4
$$

Thus, the critical point for the function $$ f(x) = -8x^2 + 64x - 2 $$ is $$ x = 4 $$.
The critical point is $$ x = 4 $$.

Find all critical points for the function . Separate multiple answers with a comma.

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