Question

For what values of x is `f(x)= -2x^3 + 4 x^2 + 5x -5` concave or convex?

Answer 1

Concave: `{x in RR: 2/3< x < oo}`

Convex: `{x in RR: -oo < x < 2/3 }`

Explanation:

A function is convex if the second derivative is positive and concave where its second derivative is negative. When the second derivative is `0` this could mean the function is concave , convex, or it could be a point of inflexion. This would have to be tested using the first derivative.

`f(x)=-2x^3+4x^2+5x-5`

`dy/dx(-2x^3+4x^2+5x-5)=-6x^2+8x+5`

`(d^2y)/(dx^2)(-6x^2+8x+5)=-12x+8`

So:

`-12x+8<0`

`x>8/12`

`x >2/3`

Concave: `{x in RR: 2/3< x < oo}`

`-12x+8>0`

`x<8/12`

`x<2/3`

Convex: `{x in RR: -oo < x < 2/3 }`

Graph:

graph{y=-2x^3+4x^2+5x-5 [-4, 4, -7.12, 7.12]}