Question

If: $f\left(x\right)=\frac{x(e^{1/x}-e^{-1/x})}{e^{1/x}+e^{-1/x}},....x\ne 0$
$=0,.....x=0$
then $f\left(x\right)$ is

• A. continuous everywhere but not differentiable at $x=0$
• B. continuous and differentiable everywhere
• C. discontinuous at $x=0$
• D. none of these

Answer 1

The correct option is B continuous and differentiable everywhere

Given, $f\left(x\right)=\frac{x(e^{1/x}-e^{-1/x})}{e^{1/x}+e^{-1/x}},....x\ne 0$
$=0,.....x=0$
$f^{\prime }\left(x\right)=\frac{(e^{1/x}-e^{-1/x})(e^{1/x}-1/x^2-e^{-1/x}.1/x^2)}{e^{1/x}.-1/x^2+e^{-1/x}.1/x^2}$
$\therefore f^{\prime }\left(x\right)$ is existed. So, we know that "Every differentiable function is a continuous function but converse need not to be true". So, the given function is continuous and differentiable everywhere.