Max, Min

Calculus Level 3

In the following functions, \(x\) is any real number.

\[\large f(x) = |x|+1 \\ \large g(x)=x^2+1\] Let \( h(x) = \begin{cases} \text{ max}\{ f(x), \ g(x) \} & \text{ if } x\le 0 \\ \text{ min}\{ f(x), \ g(x) \} & \text{ if } x>0 \end{cases} \)

Find the total number of points at which \( h(x) \) is not differentiable.

Notation: \( | \cdot | \) denotes the absolute value function.

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