Question

Find f'(x) and find the value(s) of x where the tangent line is horizontal.
f(x)=x³(x-8)⁵

Answer 1

To find f'(x) and the values of x where the tangent line is horizontal for f(x)=x³(x-8)⁵, we apply the product rule and chain rule to find f'(x), and then set f'(x) = 0 to solve for x.

Step-by-step explanation:

To find f'(x), we need to apply the product rule and the chain rule. Let's start by applying the product rule:

• Derivative of the first term: 3x⁵(x-8)ⁱ
• Derivative of the second term: x³(5(x-8)⁰)

Now, let's apply the chain rule to find the derivative of (x-8)ⁱ:

• Derivative of the outer function: (x-8)ⁱ
• Derivative of the inner function: 1

Putting it all together, we get:

f'(x) = 3x⁵(x-8)ⁱ + x³(5(x-8)⁰)(1)

To find the values of x where the tangent line is horizontal, we set f'(x) = 0 and solve for x.