Question

If $f\left(x\right)=sgn\left(x^5\right)$ then which of the following is/are false (where sgn denotes signum function)

• A. $f^{\prime }\left(0^{+}\right)=1$
• B. $f^{\prime }\left(0^{-}\right)=1$
• C. f is continuous but not differentiable at x = 0
• D. f is discontinuous at x = 0

Answer 1

Answer: (A), (B), (C)

The correct options are
A $f^{\prime }\left(0^{+}\right)=1$
B $f^{\prime }\left(0^{-}\right)=1$
C f is continuous but not differentiable at x = 0

$f\left(x\right)=sgn\left(x^5\right)=1$ if $x>0$
$f\left(x\right)=sgn\left(x^5\right)=0$ if $x=0$
$f\left(x\right)=sgn\left(x^5\right)=-1$ if $x<0$
$f^{\prime }\left(0^{+}\right)=\underset{h\to 0}{lim}\frac{1-0}{h}=DNE$
$f^{\prime }\left(0^{-}\right)=\underset{h\to 0}{lim}\frac{-1-0}{-h}=DNE$
Also, $f\left(0^{+}\right)=1\&f\left(0^{-}\right)=-1$
$\Rightarrow$ $f$ is neither continuous not differentiable at $x=0$