Question

How do you determine whether the function `f(x)=3x^5 - 20x^3` is concave up or concave down and its intervals?

Answer 1

We use the second derivative test and find that
f is concave down on `(-oo ; -sqrt2)uu(0;sqrt2)` and concave up on `(-sqrt2;0)uu(sqrt2;oo)`

Explanation:

For concavity we use the second derivative test.

`f'(x)=15x^4-60x^2`
`f''(x)=60x^3-120x =60x(x^2-2)`

This second derivative equals zero if `x = 0` or `x = - sqrt2` or `x = sqrt2`.

These are then the possible inflection points of the function where concavity could change.
We now investigate the sign of the second derivative around these points :

___ -root2 __ 0 ___root2 __
F''(x) ; - + - +

`therefore f` is concave down on `(-oo ; -sqrt2)uu(0;sqrt2)` and concave up on `(-sqrt2;0)uu(sqrt2;oo)`

graph{3x^5-20x^3 [-10, 10, -5.21, 5.21]}