Question

Let $f\left(x\right)=\left|x\right|+\left|\sin x\right|in(-\pi /2,3\pi /2)$. Then $f$ is

• A. continuous nowhere
• B. continuous everywhere
• C. differentiable nowhere
• D. differentiable everywhere except at $x=0$

Answer 1

Answer: (B), (D)

The correct options are
B continuous everywhere
D differentiable everywhere except at $x=0$

$f\left(x\right)=|x|+|sinx|x\in (-\pi /2,3\pi /2)piecewisefunctionsoff\left(x\right)aref\left(x\right)=-x-sinxx\in (-\pi /2,0]=+x+sinxx\in (0,\pi /2]=+x-sinxx\in (\pi /2,3\pi /2)f^{\prime }\left(x\right)=-1-cosxx\in (-\pi /2,0]=1+cosxx\in (0,\pi /2]=1-cosxx\in (\pi /2,3\pi /2)\underset{x\to 0^{+}}{Lt}f\left(x\right)=\underset{x\to 0^{-}}{Lt}f\left(x\right)=f\left(0\right)=0and\underset{x\to \pi /2^{+}}{Lt}f\left(x\right)=\underset{x\to \pi /2^{-}}{Lt}f\left(x\right)=f\left(\pi /2\right)=1\Rightarrow f\left(x\right)iscontinuousatx=0,\pi /2f^{\prime }\left(0^{+}\right)=0and{f}^{\prime }\left(0^{-}\right)=2f^{\prime }\left(0^{+}\right)\ne f^{\prime }\left(0^{-}\right)\Rightarrow f\left(x\right)isnotdifferentiabletox=0$