4.1 MAXIMUM AND MINIMUM VALUES THE EXTREME VALUE THEOREM If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers c and d in [a, b] Definition: Let c be a number in the domain D of a function f. Then f(c) is the: ABSOLUTE MAXIMUM value of f on D if f (c) f ( x) for all x in D ABSOLUTE MINIMUM value of f on D if f (c) f ( x) for all x in D Definition: The number f(c) is : LOCAL MAXIMUM value of f if f (c) f ( x) when x is near c. LOCAL MINIMUM value of f if f (c) f ( x) when x is near c. Critical Numbers are the x values where Find Critical Numbers f ' ( x) 0 or is undefined Find the critical numbers of the function f ( x) x 3 6 x 2 15 x f ( ) 4 tan f ( x) x 4 x 3 x 2 1 f ( x) x 1 x f ( x ) ( x 2 1)3 f ( x) x2 x2 9 Find the absolute and local maximum and minimum values of f ( x) x 3 6 x 2 5 on [3,5] f ( x) x ln x on [ 12 ,2] f ( x) 2 cos x sin 2 x on [0, ]