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For which of the following decreasing functions f does (f−1)′(10)=−1/8

Sagot :

Recall the inverse function theorem: if f(x) has an inverse, and if f(a) = b and a = f⁻¹(b), then

f⁻¹(f(x)) = x   ⇒   (f⁻¹)'(f(x)) • f'(x) = 1   ⇒   (f⁻¹)'(f(x)) = 1/f'(x)

⇒   (f⁻¹)'(b) = 1/f'(a)

Let b = 10. Then pick the function f(x) such that f(a) = 10 and f'(a) = -8 for some number a.

The decreasing function which f⁻¹'(10) = -1/8 is f(x) = -2x³ - 2x + 14

How to determine the function?

To do this, we make use of the following  inverse function theorem.

Given that f(a) = b then a = f⁻¹(b)

The above means that:

If f⁻¹(10) = 1/8

Then f(1/8) = 10

From the list of options, we have:

f(x) = -2x³ - 2x + 14

Set f(x) = 10

-2x³ - 2x + 14 = 10

Subtract 10 from both sides

-2x³ - 2x + 4 = 0

Divide through by -2

x³ + x - 2 = 0

Expand

(x - 1)(x² + x + 2) = 0

Split

x - 1 = 0 or x² + x + 2 = 0

The equation x² + x + 2 = 0 has no real solution

So, we have:

x - 1 = 0

Solve for x

x = 1

Differentiate function f(x)

f'(x) = -(6x² + 2)

Take the inverse of both sides

[tex]\frac{1}{f'(x)} = \frac{1}{-(6x\² + 2)}[/tex]

Substitute 10 for x

[tex]\frac{1}{f'(1)} = \frac{1}{-(6 * 1\² + 2)}[/tex]

[tex]\frac{1}{f'(1)} = -\frac{1}{8}[/tex]

This means that:

f⁻¹'(10) = -1/8

Hence, the function which f⁻¹'(10) = -1/8 is f(x) = -2x³ - 2x + 14

Read more about inverse functions at:

https://brainly.com/question/14391067

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