Question

$f\left(x\right)=\{\begin{array}{c}\left|\begin{array}{c}x-4\end{array}\right|forx\ge 4\\ x^3/2-x^2+3x+1/2forx<1\end{array}$ then

• A. $f\left(x\right)$ is continuous at $x=1$ and $x=4$
• B. $f\left(x\right)$ is differentiable at $x=4$
• C. $f\left(x\right)$ is continuous and differentiable at $x=1$
• D. $f\left(x\right)$ is only continuous at $x=1$

Answer 1

The correct option is A $f\left(x\right)$ is continuous at $x=1$ and $x=4$

For functions defined on open interval $(a,b)$ continuously is checked by

$1.$RHL $=f\left(a\right)$

$2.$LHL $=f\left(b\right)$

At $x=4$

${lim}_{x\to 4^{+}}f\left(x\right)={lim}_{x\to 4^{+}}(x-4)=4-4=0$

$f\left(4\right)=|4-4|=0$

RHL$=f\left(4\right)\Rightarrow$ continuous

At $x=1$

${lim}_{x\to 1^{-}}f\left(x\right)=\frac{1}{2}-1+3+\frac{1}{2}=3$

$f\left(1\right)=\frac{1}{2}-1+3+\frac{1}{2}=3$

LHL$=f\left(1\right)\Rightarrow$continuous