Download Finding Extrema on a Closed Interval

Miss Battaglia
AP Calculus AB/BC

Min & max are the largest and smallest value
that the function takes at a point
Let f be defined as an interval I containing c.


f(c) is the min of f on I if f(c)<f(x) for all x in I
f(c) is the max of f on I if f(c)>f(x) for all x in I
Not a
max
max
min
f is continuous
[-1,2] is closed
min
f is continuous
(-1,2) is open
max
Not a min
g is not continuous
[-1,2] is closed
If f is continuous on a closed interval [a,b] then
f has both a minimum and a maximum on the
interval.

Think of a relative max as occurring on a “hill” on the
graph and a relative min as occurring on a “valley” of a
graph.
If there is an open interval containing c on which f(c) is a
max, then f(c) is called a relative max of f, or you can say
f has a relative max at (c,f(c))
If there is an open interval containing c on which f(c) is a
min, then f(c) is called a relative min of f, or you can say
f has a relative min at (c,f(c))
AKA local max and local min
f’(c) = o or undefined
What is the value of the derivative at the relative max
(3,2)?
Find the value of the derivative at the relative min (0,0).

Let f be define at c. If f’(c)=0 or if f is not
differentiable at c, then c is a critical number of f.
c has to be in the domain of f, but does not have to
be in the domain of f’.


If f has a relative min or a relative max at
x=c, then c is a critical number of f.
Is the converse true? Think about y=x3.. Is 0 a
critical value? Is it a relative min or max?
To find the extrema of a continuous function f
on a closed interval [a,b], use the following
steps:
1.
2.
3.
4.
Find the critical numbers of f in (a,b)
Evaluate f at each critical number in (a,b)
Evaluate f at each endpoint of [a,b]
The least of these values is the minimum.
The greatest is the maximum.
Find the extrema of f(x)=3x4-4x3 on the interval [-1,2]
Find the extrema of f(x)=2x-3x2/3 on the interval [-1,3]
Find the extrema of f(x)=2sinx – cos2x on the
interval [0,2π]
 Read
3.1 Page 169 #11-27
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