Question

let f and g be inverse functions that are differentiable for all x. if f(-5)=7 and g(7)=3, which of the following statements must be false? i. f(3)= - 1/3 ii. f(-5)=1/3 iii. f(7)=1/3 a. i only b. ii only c. iii only d. i and iii only

Explanation:

Step1: Recall the formula for the derivative of an inverse - function

If \(y = f(x)\) and \(g(x)\) is the inverse of \(f(x)\), then \(g^{\prime}(y)=\frac{1}{f^{\prime}(x)}\) where \(y = f(x)\).

Step2: Analyze statement I

Given \(f(-5)=7\) and \(g^{\prime}(7) = 3\). By the formula \(g^{\prime}(y)=\frac{1}{f^{\prime}(x)}\) with \(y = 7\) and \(x=-5\), we have \(g^{\prime}(7)=\frac{1}{f^{\prime}(-5)}\), so \(f^{\prime}(-5)=\frac{1}{3}\), and statement I \(f^{\prime}(3)=-\frac{1}{3}\) is false.

Step3: Analyze statement II

Since \(f(-5)=7\), then by the inverse - function derivative formula \(g^{\prime}(7)=\frac{1}{f^{\prime}(-5)}\). Given \(g^{\prime}(7) = 3\), we get \(f^{\prime}(-5)=\frac{1}{3}\), so statement II \(f^{\prime}(-5)=\frac{1}{3}\) is true.

Step4: Analyze statement III

We know that if \(y = f(x)\) and \(g\) is the inverse of \(f\), \(g^{\prime}(y)=\frac{1}{f^{\prime}(x)}\). There is no information to support the relationship in statement III \(f^{\prime}(7)=\frac{1}{3}\), and it is not consistent with the inverse - function derivative formula based on the given \(f(-5)=7\) and \(g^{\prime}(7) = 3\), so statement III is false.

Answer:

A. I only