# For what values of x is  f(x) = e^(4x)-4e^(2x)+e^x  concave or convex?

Nov 20, 2017

Concave: (-∞,-2.769)∪(-0.03,+∞)
Convex: $\left(- 2.769 , - 0.03\right)$

#### Explanation:

First we need to calculate the second derivate of the function,
$f ' ' \left(x\right) = 16 {e}^{4 x} - 16 {e}^{2 x} + {e}^{x}$

then we equal it to $0$,
$16 {e}^{4 x} - 16 {e}^{2 x} + {e}^{x} = 0$

and we solve for $x$,
${x}_{1} \cong - 2.769$
${x}_{2} \cong - 0.03$

and we substitute the second derivate by a number between these intervals: (-∞,-2.769) and $\left(- 2.769 , - 0.03\right)$ and (-0.03,+∞). If the number we get is negative it means that the function is convex in that interval, if it's possitive means that's concave,
f''(-10)=4.54·10^-5
$f ' ' \left(- 1\right) = - 1.5$
$f ' ' \left(5\right) = 7762290852$