Question

let f and g be inverse functions that are differentiable for all x. if f(3)= - 2 and g(-2)= - 4, which of the following statements must be false?
i. f(0)=\frac{1}{4}
ii. f(3)= - \frac{1}{4}
iii. f(5)= - \frac{1}{4}
(a) i only
(b) ii only
(c) iii only
(d) i and iii only

Explanation:

Step1: Use the formula for the derivative of inverse functions

If \(y = f(x)\) and \(x = g(y)\) (where \(g=f^{-1}\)), then \(g^{\prime}(y)=\frac{1}{f^{\prime}(x)}\)

Step2: Analyze statement I

Since \(g(x)\) is the inverse of \(f(x)\), if \(y = f(x)\), then \(x = g(y)\). When \(y = 0\), assume \(x=a\) such that \(f(a)=0\). Also, \(g^{\prime}(y)=\frac{1}{f^{\prime}(x)}\). But we have no information about the relationship between \(x\) and \(y\) for \(y = 0\). However, if we use the formula \(g^{\prime}(y)=\frac{1}{f^{\prime}(x)}\) where \(y=f(x)\).
We know that \(g^{\prime}(y)=\frac{1}{f^{\prime}(x)}\), when \(y = f(x)\). If \(f(3)=- 2\), then \(g(-2)=3\). By the formula \(g^{\prime}(y)=\frac{1}{f^{\prime}(x)}\), \(g^{\prime}(-2)=\frac{1}{f^{\prime}(3)}\). Given \(g^{\prime}(-2)=-4\), then \(\frac{1}{f^{\prime}(3)}=-4\), so \(f^{\prime}(3)=-\frac{1}{4}\)

Step3: Analyze statement II

Since \(g^{\prime}(y)=\frac{1}{f^{\prime}(x)}\) and \(f(3)=-2\), \(g(-2) = 3\). We found \(f^{\prime}(3)=-\frac{1}{4}\) (correct)

Step4: Analyze statement III

Let \(y = f(x)\), if we want to find \(f^{\prime}(5)\), we need to know \(g\) - related value. There is no information in the problem that relates \(x = 5\) for \(f\) and \(g\). But if we assume \(y=f(x)\) and \(x = g(y)\), to use \(g^{\prime}(y)=\frac{1}{f^{\prime}(x)}\), we have no data for \(y\) such that \(x = 5\) (i.e., no \(g(y)=5\) given)

Answer:

C. III only