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

1 Answer
Nov 20, 2017

Concave: (-∞,-2.769)∪(-0.03,+∞)
Convex: (-2.769,-0.03)

Explanation:

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

then we equal it to 0,
16e^(4x)-16e^(2x)+e^x=0

and we solve for x,
x_1~=-2.769
x_2~=-0.03

and we substitute the second derivate by a number between these intervals: (-∞,-2.769) and (-2.769,-0.03) 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''(-1)=-1.5
f''(5)=7762290852