Question

The function $f$ is defined, where $x=2t-\left|t\right|,t\in R$. Draw the graph of $f$ for the interval $-1\le x\le 1$. Also discuss its continuity and differentiability at $x=0$

• A. continuous and differentiable at $x=0$
• B. not continuous but differentiable at $x=0$
• C. neither continuous nor differentiable at $x=0$
• D. differentiable but not continuous at $x=0$

Answer 1

The correct option is A continuous and differentiable at $x=0$

For $t\ge 0$
$x=2t-t=t$
$y=t^2+t^2=2t^2$
For $t<0$
$x=3t,y=0$
$0\le x<1\Rightarrow 0\le t<1$ [$\therefore x=t$]
$-1\le x<0\Rightarrow \frac{-1}{3}\le t<0$ [$\therefore x=3t$]
$f$ is continuous and differentiable at $x=0$