# What are the local extrema of f(x)= 4^x if they exist?

If $f \left(x\right) = {4}^{x}$ has a local extremum at c, then either

$f ' \left(c\right) = 0$ or $f ' \left(c\right)$ does not exist.

(The ' symbolizes the first derivative)

Hence

$f ' \left(x\right) = {4}^{x} \cdot \ln 4$

Which is always positive, so $f ' \left(x\right) > 0$ hence the function does not

have a local extrema.