Question 7.3.3: Suppose that f and g are twice differentiable functions whic......
Suppose that f and g are twice differentiable functions which are inverses of each other. By differentiating g^{\prime}( f (x)) = 1/f^{\prime}(x) w.r.t. x, find an expression for g^{\prime\prime}( f (x)) where f^{\prime}(x) ≠ 0. Do f^{\prime\prime} \text{and} g^{\prime\prime} have the same, or opposite signs?
Differentiating g^{\prime}\left(f(x)\right)=1/f^{\prime}(x)\;\mathrm{w.r.t.}\;x\;\mathrm{yields}
g^{\prime\prime}\left(f(x)\right)f^{\prime}(x)=\left(-1\right)\left(f^{\prime}(x)\right)^{-2}f^{\prime\prime}(x)It follows that, if f^{\prime}(x) \neq 0, then
g^{\prime\prime}\left(f(x)\right)=-\frac{f^{\prime\prime}(x)}{(f^{\prime}(x))^{3}} (7.3.3)
If f^{\prime}> 0, then f^{\prime\prime}(x) and g^{\prime\prime}( f (x)) have opposite signs, but they have the same sign if f^{\prime}< 0. In particular, if f is increasing and concave, the inverse g is increasing and convex, as shown in Fig. 7.3.1.
