# Math Help - need help with finding points?

1. ## need help with finding points?

Use the derivative of the function y=f(x) to find the points at which f has a local maximum, local minimum or point of inflection.

y ' = 6(x+1)(x-2)^2

help me with this problem, thank you

2. You already know that $*y' = 6(x+1)(x-2)^2$*
At an extremum, $y' = 6(x+1)(x-2)^2 = 0$
That is $x\in \{-1,2\}$.
To know the nature of these points we compute the second order derivative.
$y'' = 6(x-2)^2 + 12(x+1)(x-2)= [6(x-2)+12(x+1)](x-2) =(18x)(x-2)$
If x = -1 then $y'' =-18(-3)>0$ and we have a minimum.
If x = 2 then $y'' = 0$ and we have a point of inflection.
We also have a point of inflection at x=0 because $y'' = (18\cdot 0)(0-2)=0$.