Relative (Local) maxima are the points where the function f(x) changes its direction from increasing to decreasing.
Relative (Local) minima are the points where the function f(x) changes its direction from decreasing to increasing.
At the points of relative (local) maxima or minima, f′(x)=0.
At the points of relative (local) maxima, f′(x)<0
At the points of relative (local) minima, f′′(x)>0
Calculation: Let's say that the function is y=f(x)=2x3−21x2+36x−20. ∴f′(x)=
d
dx
f(x)=
d
dx
(2x3−21x2+36x−20)=6x2−42x+36 And, f′(x)=
d2
dx2
f(x)=
d
dx
[
d
dx
f(x)]=
d
dx
(6x2−42x+36)=12x−42 For maxima/ minima points, f′(x)=0 ⇒6x2−42x+36=0 ⇒x2−7x+6=0 ⇒x2−6x−x+6=0 ⇒x(x−6)−(x−6)=0 ⇒(x−6)(x−1)=0 ⇒x−6=0 OR x−1=0 ⇒x=6 OR x=1. Now, let's check these points for maxima/minima by inspecting the values of f′(x) at these points. f′(6)=12(6)−42=72−42=30 f′(1)=12(1)−42=12−42=−30 Since, f′(6)=30>0, it is the point of minimum value. And the minimum value is f(6): =2(6)3−21(6)2+36(6)−20 =432−756+216−20 =−128.