The interpretation of the variation table shown above is as follows: the
function F(X) increases for X in the interval (-∞, -1), reaching a maximum equal
to 36 at X = -1
Then, F(X) decreases until X
11/3, reaching a minimum of
Also, at X = ±∞, F(X)=
After that F(X) increases until reaching +∞
Using derivatives to calculate extreme points
"Extreme points," or extrema, is the general designation for maximum and
minimum values of a function in a given interval. Since the derivative of a
function at a given point represents the slope of a line tangent to the curve at
that point, then values of x for which f'(x) =0 represent points where the graph
of the function reaches a maximum or minimum.
Furthermore, the value of
the second derivative of the function, f"(x), at those points determines whether
the point is a relative or local maximum [f"(x)<0] or minimum [f"(x)>0]. These
ideas are illustrated in the figure below.
In this figure we limit ourselves to determining extreme points of the function y
= f(x) in the x-interval [a,b]. Within this interval we find two points, x = x
and x = x
, where f'(x)=0. The point x = x
, where f"(x)>0, represents a local
minimum, while the point x = x
, where f"(x)<0, represents a local maximum.
From the graph of y = f(x) it follows that the absolute maximum in the interval
[a,b] occurs at x = a, while the absolute minimum occurs at x = b.