Warm up Problem
A poster is to contain
100 in.2 of picture,
with a 4 in. margin on
top and bottom, and a 2
in. margin on left and
right. Find the overall
dimensions that will
minimize the total area
of the poster.
Miscellaneous Theorems
Thm. Extreme Value Theorem
If f (x) is continuous on a closed interval, then
it has an absolute max. and an absolute min. on
the interval.
Thm. Intermediate Value Theorem
Let f (x) be continuous on the interval [a,b]. If
k is any number between f (a) and f (b), then
there is a point c on [a,b] such that f (c) = k.
3
Every y-coordinate
between the
endpoints is hit
2
1
2
-1
4
Ex. Show that f (x) = x5 – 3x2 + 1 has a
zero on the interval [-1,2].
Why did the chicken cross the road?
[Assume the chicken’s path is a continuous function
with starting point on one side of the road and ending
point on the other side of the road.]
Thm. Mean Value Theorem
If f (x) is continuous on the interval [a,b] and
differentiable on the interval (a,b), then there is
some point c on the interval such that
f b   f  a 
f c 
ba
Ex. Let f (x) = x2 + 2x – 1. Find c on the
interval [-1,2] that satisfies MVT.
This is the last lesson from this
chapter. Next class we will review,
and your chapter test will be on
Monday