How to finding Local Maxima and Minima by Differentiation

Our example will be : $f(x)=x^{3}-12x+1$ And the graph will be the one at the side.

Then we will need to use the derivatives to find the two-point that has 0 slopes we will also call it Local maxima and Local minima

$f'(x)=3x^{2}-12$

$f'(x)=3x^{2}-12$

$3x^{2}=12$

$x^{2}=4$

Then we get x=2 and x=-2 so then we put them into the function.

$f(x)=2^{3}-(12*2)+1=-15$

$f(x)=-2^{3}-(12*-2)+1=17$

so the first point will be local maxima it will be at (-2,17) and then local minima (2,-15).