Derivatives and Concavity of an Inverse Function
Let $f$ be a function with continuous derivatives satisfying $f'(x) > 0$ and $f''(x) > 0$ for all $x$. If $g$ is the inverse function of $f$, which of the following statements about $g$ is true for all $x$ in its domain?
A
g’(x) < 0 and g’'(x) < 0 for all $x$ in the domain of $g$
B
g’(x) > 0 and g’'(x) > 0 for all $x$ in the domain of $g$
C
g’(x) < 0 and g’'(x) > 0 for all $x$ in the domain of $g$
D
g’(x) > 0 and g’'(x) < 0 for all $x$ in the domain of $g$
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