13. let f(x) = { x² + 1 if x < 1; x + 1 if x ≥ 1. is f differentiable at x = 1? sketch the graph of f.

Explanation:

Step1: Check left - hand derivative

For $x < 1$, $f(x)=x^{2}+1$. The derivative $f^\prime(x) = 2x$. The left - hand derivative at $x = 1$ is $\lim_{x\rightarrow1^{-}}f^\prime(x)=\lim_{x\rightarrow1^{-}}2x=2$.

Step2: Check right - hand derivative

For $x\geq1$, $f(x)=x + 1$. The derivative $f^\prime(x)=1$. The right - hand derivative at $x = 1$ is $\lim_{x\rightarrow1^{+}}f^\prime(x)=1$.

Step3: Compare left and right derivatives

Since the left - hand derivative $\lim_{x\rightarrow1^{-}}f^\prime(x)=2$ and the right - hand derivative $\lim_{x\rightarrow1^{+}}f^\prime(x)=1$, and $2\neq1$.

Answer:

$f(x)$ is not differentiable at $x = 1$.