Question

Inverse Function Theorem in the text says: If f is a differentiable function that is one-to-one near a and f'(a) ? 0, then 1. f?¹(x) is defined for x near b = f(a). 2. f?¹(x) is differentiable near b = f(a). 3. and last, but not least: [d/dx f?¹(x)]??_b = 1 / f'(a) where b = f(a) Explain why [d/dx f?¹(x)]??_b involves f'(a) instead of f'(b). Also explain why the condition f'(a) ? 0 is necessary.

          Inverse Function Theorem in the text says:

If f is a differentiable function that is one-to-one near a and f'(a) ? 0, then

1. f?¹(x) is defined for x near b = f(a).
2. f?¹(x) is differentiable near b = f(a).
3. and last, but not least:
[d/dx f?¹(x)]??_b = 1 / f'(a) where b = f(a)

Explain why [d/dx f?¹(x)]??_b involves f'(a) instead of f'(b). Also explain why the condition f'(a) ? 0 is necessary.

Inverse Function Theorem in the text says:

If f is a differentiable function that is one-to-one near a and f'(a) ? 0, then

1. f?¹(x) is defined for x near b = f(a).
2. f?¹(x) is differentiable near b = f(a).
3. and last, but not least:
[d/dx f?¹(x)]??b = 1 / f'(a) where b = f(a)

Explain why [d/dx f?¹(x)]??b involves f'(a) instead of f'(b). Also explain why the condition f'(a) ? 0 is necessary.

Added by Ian P.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Madhur L

06/05/2023

Step 1

We are given that f is a differentiable function that is one-to-one near a and f'(a) ≠ 0. This means that f has a unique inverse function, f^(-1)(x), near x = b = f(a). Show more…

Show all steps

Thanks for your feedback!

Inverse Function Theorem in the text says: If f is a differentiable function that is one-to-one near a and f'(a) ≠ 0, then 1. f^(-1)(x) is defined for x near b = f(a). 2. f^(-1)(x) is differentiable near b = f(a). 3. And last, but not least: f^(-1)'(b) = 1 / f'(a) where b = f(a). Explain why f^(-1)'(x) involves f'(a) instead of f'(b). Also explain why the condition f'(a) ≠ 0 is necessary: