1. Do your homework meticulously!!!
3.1 Extrema on an Interval
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Do your homework meticulously!!!

2. Definition of Extrema
f(c) is the minimum of f on I if f(c) f(x) x in I.
f(c) is the maximum of f on I if f(c) f(x) x in I.
If f is continuous on a closed interval [a,b], then f has
both a minimum and a maximum on the interval.
If f is defined at c, then c is called a critical number
of f if f’(c) = 0 or if f’(c) is undefined at c.
Relative Extrema occur only at critical numbers.
If f has a relative min or relative max at x = c, then
c is a critical number of f.

3. Ex. Find the absolute extrema of f(x) = 3x4 – 4x3
on the interval[-1, 2].
f’(x) = 12x3 – 12x2
0 = 12x3 – 12x2
0 = 12x2(x – 1)
x = 0, 1 are the critical numbers
Evaluate f at the
endpoints of [-1, 2]
and the critical #’s.
f(-1) =
f(0) =
f(1) =
f(2) = 7 -1
abs. min. 16
abs. max.

4. Ex. Find the absolute extrema of f(x) = 2x – 3x2/3
on [-1, 3]
= 0
C. N. ‘s are x = 1 because f’(1) = 0
and x = 0 because f’ is undefined
Evaluate f at the
endpoints of [-1, 3]
and the critical #’s.
f(-1) =
f(0) =
f(1) =
f(3) = -5
abs. min.
abs. max. -1
-.24

5. Ex. Find the absolute extrema of
f(x) = 2sin x – cos 2x on
f’(x) = 2cos x + 2sin 2x
sin 2x = 2 sin x cos x
0 = 2cos x + 2(2sin x cos x)
Factor -1
0 = 2cos x(1 + 2sin x) 3
max
2cos x = 0
cos x = 0 -3 2
min
sin x = -1/2 -1
Evaluate f at the
endpoints of
and the critical #’s. -3 2
min -1