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 $x = 5$.
$f(x)=\begin{cases}x^{2}-20&\text{for }x < 5\\-x^{2}+20&\text{for }xgeq5end{cases}$

Explanation:

Step1: Check continuity at x = 5

Find left - hand limit: $\lim_{x
ightarrow5^{-}}f(x)=\lim_{x
ightarrow5^{-}}(x^{2}-20)=5^{2}-20 = 5$.
Find right - hand limit: $\lim_{x
ightarrow5^{+}}f(x)=\lim_{x
ightarrow5^{+}}(-x^{2}+20)=-5^{2}+20=-5$.
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 $x = 5$.