Question

For what values of x is `f(x)=(x-3)(x+2)(x-1)` concave or convex?

Answer 1

Refer Explanation.

Explanation:

Given that: `f(x) =(x-3)(x+2)(x-1)`
`:.` `f(x) =(x^2-x-6)(x-1)`
`:.` `f(x) =(x^3-x^2-6x-x^2+x+6)`
`:.` `f(x) =(x^3-2x^2-5x+6)`

By using second derivative test,

1. For the function to be concave downward:`f''(x)<0`
   `f(x) =(x^3-2x^2-5x+6)`
   `f'(x) =3x^2-4x-5`
   `f''(x) =6x-4`
   For the function to be concave downward:
   `f''(x)<0`
   `:.` `6x-4<0`
   `:.` `3x-2<0`
   `:.` `color(blue)(x<2/3)`

2. For the function to be concave upward:`f''(x)>0`
   `f(x) =(x^3-2x^2-5x+6)`
   `f'(x) =3x^2-4x-5`
   `f''(x) =6x-4`
   For the function to be concave upward:
   `f''(x)>0`
   `:.` `6x-4>0`
   `:.` `3x-2>0`
   `:.` `color(blue)(x>2/3)`