ex: 3x4 – 16x3 + 18x2 for -1 < x < 4

3.1 – Maximums and Minimums

• Global vs. Local

• Global = highest / lowest point in the domain or interval…

• Local = high / low points relative to points NEAR a particular x value…

• Endpoints of an interval cannot be local…

Local

Global

Extreme Value Theorem

“If  f is continuous on a closed interval [a, b] then  f has a global maximum and a global minimum at some numbers in that interval.”

• Extreme values can occur more than once.• Absence of continuity or closed interval requirements negates the theorem. (An extreme value might not exist).

Fermat’s Theorem“If  f has a local maximum or minimum at c, and if f ’(c) exits, then f ’(c) = 0”• The converse of this is false in general… f’(c) = 0 does NOT automatically guarantee a local max or min…• ex: f(x) = x3, f(x) = |x|• We can extend this theorem to the idea of ‘critical numbers’.

A critical number of a function f is a number c in the domain of f such that f ’(c) = 0 OR f ’(c) does not exist.

ex: Find the critical numbers of:

Product Rule:

• Find values where the derivative = 0 or is undefined:

f’(x) = 0 where the top = 0

f’(x) = undefined where the bottom = 0

Critical numbers are 0, 3/2

Re-wording of Fermat: “If f has a local maximum or minimum at c, then c is a critical number of f…”

The Closed Interval Method

• Continuous functions on closed intervals always have a max and a min. (EVT)• Global maximums or minimums can be local or at endpoints of an interval. • If they are local, they are at critical numbers (Fermat)• So to find global max/min we test any critical numbers and both endpoints of an interval in a function and compare the results.

To find the global max/min of a continuous function f on a closed interval [a,b]:1. Evaluate f at any critical numbers in [a,b].2. Evaluate f at the endpoints of the interval.3. Compare the results – the largest value is the global max, the smallest value is the global min.

ex: Use calculus to find the exact minimum and maximum values of the following function:

1. Find critical values…

*(the derivative of f is continuous on the interval, thus there are no critical numbers where the derivative is undefined…)

2. Evaluate critical values…

3. Evaluate end points…

4. Compare

MAX

MIN