The number of points at which the function f( | vedclass

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(C) We are given the function $f(x) = \max \{a-x, a+x, b\}$.To find the points of non-differentiability, we analyze the intersections of the functions

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$2$

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(C) We are given the function $f(x) = \max \{a-x, a+x, b\}$.To find the points of non-differentiability, we analyze the intersections of the functions $y_1 = a-x$, $y_2 = a+x$, and $y_3 = b$.$1$. Intersection of $y_1$ and $y_3$: $a-x = b \implies x = a-b$.$2$. Intersection of $y_2$ and $y_3$: $a+x = b \implies x = b-a$.$3$. Intersection of $y_1$ and $y_2$: $a-x = a+x \implies 2x = 0 \implies x = 0$.Since $0  0$.At $x = a-b$, $f(x)$ transitions from $a-x$ to $b$, creating a sharp turn.At $x = b-a$, $f(x)$ transitions from $b$ to $a+x$, creating a sharp turn.At $x = 0$, the value of $f(0) = \max \{a, a, b\} = b$ (since $b > a$). The function is $b$ in the interval $[a-b, b-a]$, so it is differentiable at $x=0$.Thus, there are exactly $2$ points of non-differentiability: $x = a-b$ and $x = b-a$.

The number of points at which the function $f(x) = \max \{a-x, a+x, b\}$ for $-\infty < x < \infty$ and $0 < a < b$ is not differentiable, is

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