Question

What are the values and types of the critical points, if any, of `f(x) =x^3 + 3x^2-24x`?

Answer 1

`(2, -28)` Minimum Point,
`(-4, 80)` Maximum Point and `(-1, 26)` Point of inflection

Explanation:

`f(x)=x^3+3x^2-24x` the given
`f' (x)=3 x^2 +6 x-24` the first derivative

Let `f' (x) = 0`
`3 x^2 +6 x-24=0`
solving for x:
`x=2` and `x=-4`
when `x=2` , `y=-28` Minimum point
when `x=-4`, `y=80` Maximum Point

Solving for `f'' (x)`
`f'' (x)=6x+6`
set `f''(x)=0` and solving for x:
`6x+6=0`
`x=-1`
when `x=-1`, `y=26` Point of inflection