Section 4.1 – Extreme Values of Functions

• Using the first derivative to find maximum and minimum values (max. and min.)

• Find critical points to locate these values

Absolute (Global) Extreme Values

• Definition – Absolute Extreme ValuesLet f be a function with domain D. Then f(c) is

the(a) absolute maximum value on D if and only if

f(x) ≤ f(c) for all x in D.(b) absolute minimum value on D if and only if

f(x) ≥ f(c) for all x in D.

Theorem 1 – The Extreme Value Theorem

• If f is continuous on a closed interval [a,b], then f has both a maximum value and a minimum value on the interval.

Example 1 (y = x2)

• y = x2 , (- , )

• No absolute maximum• Absolute minimum

of 0 at x = 0

Example 2 (y = x2)

• y = x2 , [0,2]

• Absolute maximum of 4 at x = 2

• Absolute minimumof 0 at x = 0

Example 3 (y = x2)

• y = x2 , (0,2]

• Absolute maximumof 4 at x = 2

• No absolute minimum

Example 4 (y = x2)

• y = x2 , (0,2)

• No absolute extrema

Possible locations…

Possible locations…(2)

Absolute vs. Local

Local Extreme Values

• Definition – Local Extreme ValuesLet c be an interior point of the domain of the

function f. Then f(c) is a (a) local maximum value at c if and only if

f(x) ≤ f(c) for all x in some open interval containing c.

(b) local minimum value at c if and only iff(x) ≥ f(c) for all x in some open interval containing c.

Theorem 2 – Local Extreme Values

• If a function f has a local maximum value or a local minimum value at an interior point c of its domain, and if f’ exists at c, then

f’(c) = 0• Definition – Critical Point

A point in the interior of the domain of a function f at which f’ = 0 or f’ does not exist is a critical point of f.

Bellringer

Find the derivative of x3 – x2 – 3x + 1

Example 1 (Graphically)

f(x) = x3 – x2 – 3x + 1f’(x) = x2 – 2x – 3

D: [-4, 6]

Example 1 (Analytically)

f(x) = x3 – x2 – 3x + 1 D: [-4, 6]f’(x) = x2 – 2x – 3

x2 – 2x – 3 = 0 f(-4) = -24.333(x - 3)(x + 1) = 0 f(-1) = 2.667x = -1, 3 f(3) = -8

f(6) = 19

Example 2

f(x) =

What is our domain?Why?f’(x) = We already know our function is undefined at certain

pointsCritical point is at x = 0f(0) =

Steps to check

1. Is the function undefined or 0 at any points?2. Find the derivative and find any places the

derivative is not defined or is equal to 0.3. Use these values in f(x) and find their values.

Check the domain values in f(x) as well.4. Determine if they are max. or min. values

(Global or Local)

Example 3

f(x) = ln | |Undefined?f‘(x) = Critical points?Check f(x) values?Maximums or minimums?

Further Examples

• Pg. 184 (1, 3, 5, 19, 21)