f has an absolute maximum (=global maximum) at c if f(c) ≥ f(x) for all x in domain of f. f(c) is the maximum value of f. f has an absolute minimum (=global minimum) at d if f(d) ≤ f(x) for all x in domain of f. f(d) is the minimum value of f. f(c) and f(d) are the extreme values of f. f has a local maximum at c if f(c) ≥ f(x) for all x near c. f has a local minimum at d if f(d) ≤ f(x) for all x near c. Ex 1:

Ex 2: f(x) = 3 c is a critical point of f if f′(c) = 0 or if f′(c) DNE.

If a local max or local min occurs at c, then c is a critical point of f (Note this is not an iff. I.e., the converse is not true).

Extreme Value Thm: If f continuous on [a, b], then f has an absolute maximum, f(c), and an absolute minimum, f(d), for some c, d ∈ [a, b].

Extreme values must occur either at critical values (f′(x) = 0 or DNE) or at endpoints.

Ex: Let f(x) = x^{3}. Find absolute minimum/maximum on the interval [-2,3].

Ex: Let f(x) = x^{3}. Find absolute minimum/maximum on the interval (-2,3) (if they exist).

Ex: Let f(x) = \{ \begin{matrix}\frac{1}{x} & x ≠ 0 \\ 0 & 0 \end{matrix} . Find absolute minimum/maximum

on the interval [-2,3] (if they exist).

Ex: Let f(x) = (x − 2)^{\frac{2}{3}} . Find absolute minimum/maximum on the interval [1,5].

Ex: Let g(x) = x^{3}+3x^{2}+1. Find absolute minimum/maximum on the interval [-3,1].

Fermat’s Thm: If a local max or local min occurs at c, then c is a critical point of f.

Proof: