James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 4.2 The Mean Value Theorem 291
Therefore f has the same value at any two numbers x 1 and x 2 in sa, bd. This means that
f is constant on sa, bd.
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Corollary 7 says that if two functions
have the same derivatives on an interval,
then their graphs must be vertical
translations of each other there. In
other words, the graphs have the same
shape, but could be shifted up or down.
7 Corollary If f 9sxd − t9sxd for all x in an interval sa, bd, then f 2 t is constant
on sa, bd; that is, f sxd − tsxd 1 c where c is a constant.
Proof Let Fsxd − f sxd 2 tsxd. Then
F9sxd − f 9sxd 2 t9sxd − 0
for all x in sa, bd. Thus, by Theorem 5, F is constant; that is, f 2 t is constant.
NOTE Care must be taken in applying Theorem 5. Let
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f sxd −
x
| x | − H 1 21
if x . 0
if x , 0
The domain of f is D − hx | x ± 0j and f 9sxd − 0 for all x in D. But f is obviously not
a constant function. This does not contradict Theorem 5 because D is not an interval.
Notice that f is constant on the interval s0, `d and also on the interval s2`, 0d.
ExamplE 6 Prove the identity tan 21 x 1 cot 21 x − y2.
SOLUtion Although calculus isn’t needed to prove this identity, the proof using calculus
is quite simple. If f sxd − tan 21 x 1 cot 21 x, then
f 9sxd − 1
1 1 x 2 1
2 1 1 x − 0 2
for all values of x. Therefore f sxd − C, a constant. To determine the value of C, we put
x − 1 [because we can evaluate f s1d exactly]. Then
C − f s1d − tan 21 1 1 cot 21 1 − 4 1 4 − 2
Thus tan 21 x 1 cot 21 x − y2.
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1. The graph of a function f is shown. Verify that f satisfies the
hypotheses of Rolle’s Theorem on the interval f0, 8g. Then
estimate the value(s) of c that satisfy the conclusion of Rolle’s
Theorem on that interval.
y
1
0
1
y=ƒ
2. Draw the graph of a function defined on f0, 8g such that
f s0d − f s8d − 3 and the function does not satisfy the
conclusion of Rolle’s Theorem on f0, 8g.
x
3. The graph of a function t is shown.
y
1
0
1
y=©
(a) Verify that t satisfies the hypotheses of the Mean Value
Theorem on the interval f0, 8g.
(b) Estimate the value(s) of c that satisfy the conclusion of the
Mean Value Theorem on the interval f0, 8g.
(c) Estimate the value(s) of c that satisfy the conclusion of the
Mean Value Theorem on the interval f2, 6g.
x
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