Calculus- Early Transcendentals, 2021a

172 Applications of Derivatives
5.2.2 Absolute Extrema
Absolute extrema are also commonly referred to as global extrema. Unlike local extrema, which are only
“extreme" relative to points “close to” them, an absolute (or global) extrema is “extreme" relative to all
other points in the interval under consideration.
Definition 5.12: Absolute Maxima and Minima
A real-valued function f has an absolute maximum on an interval I at x 0 if f (x 0 ) is the largest
value of f on I;inotherwords, f (x 0 ) ≥ f (x) for all x in the domain of f that are in I.
A real-valued function f has an absolute minimum on an interval I at x 0 if f (x 0 ) is the smallest
value of f on I;inotherwords, f (x 0 ) ≤ f (x) for all x in the domain of f that are in I.
Example 5.13: Absolute Extrema
Consider the function f (x)=x 2 on the interval (−∞,∞). This parabola has an absolute minimum
at x = 0. However, it does not have an absolute maximum.
Consider the function f (x)=|x| on the interval [−1,2]. This graph looks like a check mark. It has
an absolute minimum at x = 0 and an absolute maximum at x = 2.
Consider the function f (x)=cosx on the interval [0,π]. It has an absolute minimum at x = π and
an absolute maximum at x = 0.
Consider the function f (x)=e x on any interval [a,b],wherea < b. Since this exponential function
is increasing, it has an absolute minimum at x = a and an absolute maximum at x = b.
Like Fermat’s Theorem, the following theorem has an intuitive appeal. However, unlike Fermat’s
Theorem, the proof relies on a more advanced concept called compactness, which will only be covered
in a course typically entitled Analysis. So, we will be content with understanding the statement of the
theorem.
Theorem 5.14: Extreme-Value Theorem
If a function f is continuous on a closed interval [a,b], then f has both an absolute maximum and
an absolute minimum on [a,b].
Although this theorem tells us that an absolute extremum exists, it does not tell us what it is or how to
find it.
Note that if an absolute extremum is inside the interval (i.e. not an endpoint), then it must also be a
local extremum. This immediately tells us that to find the absolute extrema of a function on an interval,
we need only examine the local extrema inside the interval, and the endpoints of the interval.
We can devise a method for finding absolute extrema for a function f on a closed interval [a,b]:
1. Verify the function is continuous on [a,b].
2. Find the derivative and determine all critical values of f that are in [a,b].