Let $f: R \to R$ and $g: R \to R$ be respectively given by $f(x) = |x| + 1$ and $g(x) = x^{2} + 1.$ Define $h: R \to R$ by $h(x) = \max\{f(x), g(x)\}$ if $x \le 0,$ and $h(x) = \min\{f(x), g(x)\}$ if $x > 0.$ The number of points at which h(x) is not differentiable is ____.

Let f:R → R and g:R → R be respectively given by f(x)=|x|+1 and g(x)=x²+1. Define h:R → R by h(x)= max{f(x),g(x)} if x ≤ 0, min{f(x),g(x)} if x > 0. The number of points at which h(x) is not differentiable is ______. $f(x) = |x| + 1, \quad g(x) = x^2 + 1$ $h(x) = \begin{cases} \max(f,g), & x \le 0 \\ \min(f,g), & x > 0 \end{cases}$ $\text{Find: number of points where } h \text{ is not differentiable}$ $f(x) = g(x) \Rightarrow |x| = x^2 \Rightarrow x = -1, 0, 1$ $g - f = x^2 - |x| = \begin{cases} x(x+1), & x \le 0 \\ x(x-1), & x \ge 0 \end{cases}$ $h(x) = \begin{cases} x^2+1, & x \le -1 \\ 1-x, & -1 \le x \le 0 \\ x^2+1, & 0 < x \le 1 \\ x+1, & x \ge 1 \end{cases}$ $h \text{ is smooth inside each piece; check the junctions } x = -1, 0, 1$ $x = -1: \ h'_-(-1) = 2x\big|_{-1} = -2, \quad h'_+(-1) = -1$ $-2 \neq -1 \Rightarrow \text{not differentiable}$ $x = 0: \ h'_-(0) = -1, \quad h'_+(0) = 2x\big|_0 = 0$ $-1 \neq 0 \Rightarrow \text{not differentiable}$ $x = 1: \ h'_-(1) = 2x\big|_1 = 2, \quad h'_+(1) = 1$ $2 \neq 1 \Rightarrow \text{not differentiable}$ $h \text{ not differentiable at } x = -1, 0, 1 \Rightarrow \text{number} = 3$