Question

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

Answer 1

concave `(-oo,4/3)`
convex`(4/3,+oo)`

Explanation:

To determine where f(x) is concave/convex we require to find f''(x)

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

f'(x)`=3x^2-8x+5`

and f''(x)`=6x-8`

We now equate f''(x) to zero to find values of x where any change from concave/convex or convex/concave may occur.

solve : 6x - 8 = 0 `rArr x=4/3`

We now have to check the value of f''(x) to the left and right of `x=4/3 , "say " x=a`

• If f''(a) > 0 , then f(x) is convex

• If f''(a) < 0 , then f(x) is concave

x = 0 is to the left and f''(0) = - 8 → concave

x = 2 is to the right and f''(2) = 4 → convex

`"hence" f(x)" is concave " (-oo,4/3)`

and f(x)`" is convex " (4/3,+oo)`
graph{x^3-4x^2+5x [-10, 10, -5, 5]}