Ma 113 Functions And Inverse Functions, The Exponential Function And The Logarithm Worksheet Page 24
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41Worksheet # 18: Extreme Values and the Mean Value Theorem
1. Comprehension check:
(a) True or False: If f (c) = 0 then f has a local maximum or local minimum at c.
(b) True or False: If f is differentiable and has a local maximum or minimum at x = c then f (c) = 0.
(c) A function continuous on an open interval may not have an absolute minimum or absolute maximum on
that interval. Give an example of continuous function on (0, 1) which has no absolute maximum.
(d) True or False: If f is differentiable on the open interval (a, b), continuous on the closed interval [a, b], and
f (x) = 0 for all x in (a, b), then we have f (a) = f (b).
2. (a) Define the following terms or concepts:
Critical point
f has a local maximum at x = a
Absolute maximum
(b) State the following:
The First Derivative Test for Critical Points
The Mean Value Theorem
3. Sketch the following:
(a) The graph of a function defined on (
,
) with three local maxima, two local minima, and no absolute
minima.
(b) The graph of a continuous function with a local maximum at x = 1 but which is not differentiable at
x = 1.
(c) The graph of a function on [ 1, 1) which has a local maximum but not an absolute maximum.
(d) The graph of a function on [ 1, 1] which has a local maximum but not an absolute maximum.
(e) The graph of a discontinuous function defined on [ 1, 1] which has both an absolute minimum and absolute
maximum.
4. Find the critical points for the following functions:
(a) f (x) = x + x + 1
(b) g(x) = e (x
7)
(c) h(x) = 5x
1
5. Find the absolute maximum and absolute minimum values of the following functions on the given intervals.
Specify the x-values where these extrema occur.
(a) f (x) = 2x
3x
12x + 1, [ 2, 3]
(b) h(x) = x +
1
x , [ 1, 1]
6. (a) Consider the function f (x) = 2x
9x
24x + 5 on (
,
).
i. Find the critical point(s) of f (x).
ii. Find the intervals on which f (x) is increasing or decreasing.
iii. Find the local extrema of f (x).
x
(b) Repeat with the function f (x) =
on (
,
).
x + 4
π
π
(c) Repeat with the function f (x) = sin (x)
cos(x) on
,
.
2
2
7. Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find
all numbers c that satisfy the conclusion of the Mean Value Theorem.
x
(a) f (x) =
on the interval [1, 4]
x + 2
(b) f (x) = sin(x)
cos(x) on the interval [0, 2π]
8. Use the mean value theorem to show that sin(x)
x for x
0. What can you say for x
0?
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