If both f(x) and g(x) are differentiable functions at x=x0,...

If both $f(x)$ and $g(x)$ are differentiable functions at $x=x_{0}$, then the function defined as, $h(x)=$ maximum $\{f(x), g(x)\}$ (a) is always differentiable at $x=x_{0}$ (b) is never differentiable at $x=x_{0}$ (c) is differentiable at $x=x_{0}$ if $f\left(x_{0}\right) \neq g\left(x_{0}\right)$ (d) cannot be differentiable at $x=x_{0}$ if $f\left(x_{0}\right)=g\left(x_{0}\right)$

If both $f(x)$ and $g(x)$ are differentiable functions at $x=x_{0}$, then the function defined as, $h(x)=$ maximum $\{f(x), g(x)\}$
(a) is always differentiable at $x=x_{0}$
(b) is never differentiable at $x=x_{0}$
(c) is differentiable at $x=x_{0}$ if $f\left(x_{0}\right) \neq g\left(x_{0}\right)$
(d) cannot be differentiable at $x=x_{0}$ if $f\left(x_{0}\right)=g\left(x_{0}\right)$

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Key Concepts

Maximum Function in Analysis

The maximum function is a specific example of a piecewise defined function where, at each point, the function takes the value of the larger of two functions. While both component functions may be differentiable, the maximum function can exhibit corners or cusp points at the switching point where the two functions are equal, potentially causing a failure of differentiability if their derivatives do not match.

Piecewise Defined Functions

Piecewise defined functions are functions that are defined by different expressions over different intervals of the domain. When a function is defined as the maximum of two functions, it behaves piecewise, taking one of the function values depending on which is larger. The points where the definitions switch must be checked for continuity and smoothness to determine overall differentiability.

Differentiability

Differentiability is the property of a function having a derivative at a point, meaning that the function can be well approximated by a linear function near that point. In the context of piecewise-defined functions or functions defined by operations like maximum, ensuring that the left-hand and right-hand derivatives at a junction point are equal is crucial for the function to be differentiable at that point.

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