Answered

Connect with experts and get insightful answers on IDNLearn.com. Our experts provide timely, comprehensive responses to ensure you have the information you need.

The function f(x) is an increasing function about which little else is known about other than f(2)=7 and f'(2)=5. Find (f^-1)'(7).

Sagot :

LammettHashidn

Since f(x) is (strictly) increasing, we know that it is one-to-one and has an inverse f^(-1)(x). Then we can apply the inverse function theorem. Suppose f(a) = b and a = f^(-1)(b). By definition of inverse function, we have

f^(-1)(f(x)) = x

Differentiating with the chain rule gives

(f^(-1))'(f(x)) f'(x) = 1

so that

(f^(-1))'(f(x)) = 1/f'(x)

Let x = a; then

(f^(-1))'(f(a)) = 1/f'(a)

(f^(-1))'(b) = 1/f'(a)

In particular, we take a = 2 and b = 7; then

(f^(-1))'(7) = 1/f'(2) = 1/5

Thank you for using this platform to share and learn. Don't hesitate to keep asking and answering. We value every contribution you make. Thank you for trusting IDNLearn.com. We’re dedicated to providing accurate answers, so visit us again for more solutions.