Question

Let $f\left(x\right)=\{\begin{array}{c}\begin{array}{ccc}1& \forall & x<0\\ 1+\sin x& \forall & 0\le x\le \frac{\pi }{2}\end{array}\end{array}$ then what is the value of $f^{\prime }\left(x\right)$ at $x=0?$

• A. $1$
• B. $-1$
• C. $\infty$
• D. Does not exist

Answer 1

The correct option is D Does not exist

For $x<0$ , $f\left(x\right)=1$

$\Rightarrow f^1\left(0^{-}\right)=0$

For $x\ge 0$ , $f\left(x\right)=1+sinx$

$\Rightarrow f^1\left(0^{+}\right)=cos\left(0\right)=1$

Therefore $f^1\left(0^{-}\right)\ne f^1\left(0^{+}\right)$

So $f^1\left(x\right)$ at $x=0$ does not exist

Therefore the correct option is $D$