Question

let (f) be the function defined above. which of the following statements is true?
a (f) is not differentiable at (x = 5) because (f) is not continuous at (x = 5).
b (f) is not differentiable at (x = 5) because the graph of (f) has a sharp corner at (x = 5).
c (f) is not differentiable at (x = 5) because the graph of (f) has a vertical tangent at (x = 5).
d (f) is not differentiable at (x = 5) because (f) is not defined at (x = 5).
(f(x)=\begin{cases}2&\text{for }x < 5\\2x - 4&\text{for }xgeq5end{cases})

Explanation:

Step1: Check continuity at $x = 5$

Left - hand limit: $\lim_{x
ightarrow5^{-}}f(x)=2$. Right - hand limit: $\lim_{x
ightarrow5^{+}}f(x)=2\times5 - 4=6$. Since $\lim_{x
ightarrow5^{-}}f(x)
eq\lim_{x
ightarrow5^{+}}f(x)$, the function is not continuous at $x = 5$. A function must be continuous at a point to be differentiable there.

Answer:

A. $f$ is not differentiable at $x = 5$ because $f$ is not continuous at $x = 5$.