Question

For what values of x is `f(x)= -5x^3+2x^2-3x-12` concave or convex?

Answer 1

`f(x)=-5x^3+2x^2-3x-12` , `D_f=RR`

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

`f''(x)=-30x+4`

`f''(x)=0 <=> x=2/15`

For `x<2/15` , `f''(x)>0` so `f` is convex at `(-oo,2/15]`
For `x>2/15` , `f''(x)<0` so `f` is concave at `[2/15,+oo)`