Question

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

Answer 1

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`