Question

The function f(x) = e^|x| is
(a) continuous every where but not differentiable at x = 0
(b) continuous and differentiable everywhere
(c) not continuous at x = 0
(d) none of these

Answer 1

The given function is f(x) = e^|x|.

We know

If f is continuous on its domain D, then $\left|f\right|$ is also continuous on D.

Now, the identity function p(x) = x is continuous everywhere.

So, g(x) = $\left|p\left(x\right)\right|=\left|x\right|$ is also continuous everywhere.

Also, the exponential function a^x, a > 0 is continuous everywhere.

So, h(x) = e^x is continuous everywhere.

The composition of two continuous functions is continuous everywhere.

$\therefore f\left(x\right)=\left(hog\right)\left(x\right)=e^{\left|x\right|}$ is continuous everywhere.

Now,

$g\left(x\right)=\left|x\right|=\left\{\begin{array}{ll}x,& x\ge 0\\ -x,& x<0\end{array}\right.$

$Lg\text{'}\left(0\right)=\underset{h\to 0}{\lim }\frac{g\left(0-h\right)-g\left(0\right)}{-h}$

$\Rightarrow Lg\text{'}\left(0\right)=\underset{h\to 0}{\lim }\frac{-\left(-h\right)-0}{-h}$

$\Rightarrow Lg\text{'}\left(0\right)=\underset{h\to 0}{\lim }\frac{h}{-h}$

$\Rightarrow Lg\text{'}\left(0\right)=-1$

And

$Rg\text{'}\left(0\right)=\underset{h\to 0}{\lim }\frac{g\left(0+h\right)-g\left(0\right)}{h}$

$\Rightarrow Rg\text{'}\left(0\right)=\underset{h\to 0}{\lim }\frac{h-0}{h}$

$\Rightarrow Rg\text{'}\left(0\right)=\underset{h\to 0}{\lim }\frac{h}{h}$

$\Rightarrow Rg\text{'}\left(0\right)=1$

$\therefore Lg\text{'}\left(0\right)\ne Rg\text{'}\left(0\right)$

So, $g\left(x\right)=\left|x\right|$ is not differentiable at x = 0.

We know

The exponential function a^x, a > 0 is differentiable everywhere.

So, h(x) = e^x is differentiable everywhere.

We know that, the composition of differentiable functions is differentiable.

Now, e^x is differentiable everywhere, but $\left|x\right|$ is not differentiable at x = 0.

$\therefore f\left(x\right)=\left(hog\right)\left(x\right)=e^{\left|x\right|}$ is differentiable everywhere except at x = 0.

Thus, the function f(x) = e^|x| is continuous every where but not differentiable at x = 0.

Hence, the correct answer is option (a).