Question

For what values of x is `f(x)= x^4-3x^3-4x-7` concave or convex?

Answer 1

the function is convex in `]-oo;0[uu]2/3;+oo[` and concave in `]0;2/3[`

Explanation:

You would analyze the second derivative; the first one is:

`f'(x)=4x^3-9x^2-4`

then the second one is:

`f''(x)=12x^2-18x`

Let's solve the inequality `f''(x)>0`:

`12x^2-8x>0`

that's

`x<0 or x>2/3`

Then the given function is convex in

`]-oo;0[uu]2/3;+oo[`

and concave in

`]0;2/3[`

graph{x^4-3x^3-4x-7 [-5, 5, -27, 10]}