12.5: Absolute Maxima and Minima

Finding the absolute maximum or minimum

value of a function is one of the most

important uses of the derivative.

For example:

Finding the maximum profit, the time it takes

for a drug to reach its maximum concentration

in the bloodstream after an injection, or the

minimum pollution in some areas.

Absolute Maxima and Minima

Definition:

f (c) is an absolute maximum of f if f (c) > f (x) for all x in the domain of f.

f (c) is an absolute minimum of f if f (c) < f (x) for all x in the domain of f.

Local Extrema and Absolute Extrema

Extreme Value Theorem

Theorem 1. (Extreme Value Theorem)

A function f that is continuous on a closed interval [a, b] has both an absolute maximum value and an absolute minimum value on that interval.

Finding Absolute Maximum and Minimum Values

Theorem 2. Absolute extrema (if they exist) must always occur at critical values of the derivative, or at end points.

a. Check to make sure f is continuous over [a, b] .

b. Find the critical values in the interval [a, b].

c. Evaluate f at the end points a and b and at the critical values found in step b.

d. The absolute maximum on [a, b] is the largest of the values found in step c.

e. The absolute minimum on [a, b] is the smallest of the values found in step c.

Example 1

Find the absolute maximum and minimum values of f(x) = x3 – 12xA) [-5,5]

This function is continuous for all xf’(x) = 3x2 – 12 = 0 3(x-2)(x+2) = 0 so x = 2 or -2. Both are critical values.Check: f(2), f(-2), f(-5) and f(5), we have

F(2) = -16F(-2) = 16F(-5) = -65F(5) = 65

Therefore, the absolute minimum value is (-5, -65) and the absolute maximum value is (5,65)

Example 1 (continue)

Find the absolute maximum and minimum values of f(x) = x3 – 12x

B) [-3,3]

Same critical values

Check: f(2), f(-2), f(-3) and f(3), we have

F(2) = -16

F(-2) = 16

F(-3) = 9

F(3) = -9

Therefore, the absolute minimum value is (2, -16) and the absolute maximum value is (2,16)

Example 1 (continue)

Find the absolute maximum and minimum values of f(x) = x3 – 12x

C) [-3,1]

Same critical values -2 and 2, but since the interval is [-3, 1] so we do

not check f(2)

Check: f(-2), f(-3) and f(1), we have

F(-2) = 16

F(-3) = 9

F(1) = -11

Therefore, the absolute minimum value is (1, -11) and the absolute maximum value is (2,16)

Functions with no absolute extrema

• To find the local extrema and or absolute extrema of interval that is not closed, we need to sketch the graph or use the second derivative.

Important Note: Need to find out the critical values first before we can apply the 2nd derivative test

Example 2

Use the 2nd derivative test if necessary to find the local maxima and

minima for each function.

A) f(x) = x3 – 9x2 + 24x -10

f’(x) = 3x2 – 18x + 24 = 0

3(x2 – 6x + 8) = 0

3(x – 4) (x -2) = 0 so x = 2 or 4 both are critical points

f’’(x) = 6x – 18

f’’(2) = -6 < 0 so f has a local maximum at x = 2

f’’(4) = 6 > 0 so f has a local minimum at x = 4

Example 2 (continue)

Use the 2nd derivative test if necessary to find he local maxima and

minima for each function.

B) f(x) = ex – 5x

f’(x) = ex – 5 = 0

ex = 5

ln ex = ln 5 so x = ln 5 is a critical point

f’’(x) = ex

f’’(ln5) = eln5 = 5 > 0 so f has a local minimum at x = ln5

Example 2 (continue)Use the 2nd derivative test if necessary to find he local maxima and

minima for each function.

C) f(x) = 10x6 – 24x5 + 15x4

f’(x) = 60x5 – 120x4 + 60x3 = 0

60x3 (x2 – 2x + 1) = 0

60x3(x – 1)2 = 0 so x = 0 or 1 both are critical points

f’’(x) = 300x4 – 480x3 + 180x2

f’’(0) = 0 can’t determine, the 2nd derivative test is failed because f’’=0

f’’(1) = 0 can’t determine

If this occurs, use the 1st derivative test

x -1 0 .5 1 2

F’ - 0 + 0 +decreasing increasing increasing

Therefore, there is a local minimum at x=0

Second Derivative Test

Theorem 3. Let f be continuous on interval I with only one critical value c in I.

If f ’(c) = 0 and f ’’ (c) > 0, then f (c) is the absolute minimum of f on I.

If f ’(c) = 0 and f ’’ (c) < 0, then f (c) is the absolute maximum of f on I.

Example 3

A) Find the absolute maximum value of f(x) = 12 – x – 9/x on the interval (0, ∞)

F’(x) = -1+ 9/x2 = 0

- x2 + 9 = 0 multiply both sides by x2

x2 – 9 = 0, (x-3) (x+3) = 0, so x = -3 or 3, only 3 is a critical value

F’’(x) = -18/x3

F’’(3) = -2/3 < 0

Since we only have 1 critical value and f’’ < 0 so this function has an

absolute maximum at x = 3

Example 3 (continue)

B) Find the absolute maximum value of f(x) = 5lnx - x on the interval (0, ∞)

F’(x) = 5/x - 1 = 0

5/x = 1

x = 5 is a critical value

F’’(x) = -5/x2

F’’(5) = -1/5 < 0

Since we only have 1 critical value and f’’ < 0 so this function has an

absolute maximum at x = 5