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