Is this function differentiable $f(x)= \frac{e^{-|x|}}{\max (e^{x},e^{-x})}$?

Question

Consider the function $$f(x)= \frac{e^{-|x|}}{\max (e^{x},e^{-x})}$$

Is this function differentiable at every point? My progress - I was able to split the function in two parts

For $$x>0, f(x) = e^{-2x}$$

For

$$ x<0, f(x) = e^{2x}$$ Then I drew the graph and found one " pointy point" on which the function has two tangents thus I thought it is not differentiable at just one point but turns out the answer is the function is not differentiable anywhere. can you help me?

The function is just $f(x) = e^{-2|x|}$ which is differentiable everywhere except at $x=0$. — copper.hat, Jan 31 '20 at 06:55

score 2 · Answer 1 · answered Jan 31 '20 at 06:57

2

Then $f(x)=e^{-2x}. x>0; f(x)=e^{2x}, x<0, f(0)=1$ So this function is continuous but not differentiable at $x=0$ because at this point right derivative is $-2$ and the left derivative is $2$, which are finite but un-equal.

answered Jan 31 '20 at 06:57

Z Ahmed

43,235

Is this function differentiable $f(x)= \frac{e^{-|x|}}{\max (e^{x},e^{-x})}$?

1 Answers1