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(B) Given $f(x) = |x| + 1$ and $g(x) = x^2 + 1$.For $x \leq 0$,$f(x) = -x + 1$ and $g(x) = x^2 + 1$. We define $h(x) = \max\{-x+1, x^2+1\}$.Since $x^2

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$2$

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(B) Given $f(x) = |x| + 1$ and $g(x) = x^2 + 1$.For $x \leq 0$,$f(x) = -x + 1$ and $g(x) = x^2 + 1$. We define $h(x) = \max\{-x+1, x^2+1\}$.Since $x^2+1 \geq -x+1$ for $x \in [-1, 0]$ (as $x^2+x \geq 0$),$h(x) = x^2+1$ for $x \in [-1, 0]$ and $h(x) = -x+1$ for $x For $x > 0$,$f(x) = x + 1$ and $g(x) = x^2 + 1$. We define $h(x) = \min\{x+1, x^2+1\}$.Since $x^2+1 \leq x+1$ for $x \in [0, 1]$ (as $x^2-x \leq 0$),$h(x) = x^2+1$ for $x \in [0, 1]$ and $h(x) = x+1$ for $x > 1$.Thus,$h(x) = \begin{cases} -x+1 & x  1 \end{cases}$.Checking differentiability:At $x = -1$: $h(-1) = 2$. Left derivative is $-1$,right derivative is $2(-1) = -2$. Not differentiable.At $x = 1$: $h(1) = 2$. Left derivative is $2(1) = 2$,right derivative is $1$. Not differentiable.At $x = 0$: $h(0) = 1$. Left derivative is $2(0) = 0$,right derivative is $2(0) = 0$. Differentiable.There are $2$ points of non-differentiability.

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