Question

lf $f\left(x\right)=\{\begin{array}{cc}x\log \cos x;for& x\ne 0\\ 0;for& x=0\end{array}$ then,

$f^{\prime }\left(0\right)$ is __

• A. $0$
• B. $\frac{-1}{2}$
• C. $\frac{1}{2}$
• D. $1$

Answer 1

The correct option is A $0$

$f\left(x\right)=xlog\left(cosx\right);forx\ne 0$

$f\left(x\right)=0;forx=0$

$f\left(x\right)$ is continuos at $x=0$

So, $f^{\prime }\left(x\right)=\frac{x}{cosx}(-sinx)+log\left(cosx\right)$

$f^{\prime }\left(x\right)=\frac{-xsinx}{cosx}+log\left(cosx\right)$

$f^{\prime }\left(0\right)=0+log\left(cos0\right)$

$=log\left(1\right)$

$=0$

So, $f^{\prime }\left(0\right)=0.$