Question

What are the local extrema of `f(x)= x^3-3x^2-9x+7`?

Answer 1

Local maximum : `f(-1)=12`. Local minimum :`f(3)=-20`.

Explanation:

`f = x^3(1-3/x-9/x^2+7/x^3) to +-oo`, as `x to +-oo`.

f'=3(x^2-2x-3)=0, at x = -1 and 3.

`f''=6x-6, <9`, at `x = -1, >0`, at `x = 3 and = 0`, at `x =1.`

So, `local-max f = f(-1)=12 and local-min f = f(3)=-20`.

As, `f''' ne 0, ( 1, -4 )` is a POI ( point of inflexion ).

graph{(x^3-3x^2-9x+7-y)((x-1)^2+(y+4)^2-.01)=0 [-34, 34, -21, 13]}