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If h < 0, then
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But by hypothesis f x has a derivative at all points Fig. 4-9 in a; b . Then the right-hand derivative (1) must be equal to the left-hand derivative (2). This can happen only if they are both equal to zero, in which case f 0  0 as required. A similar argument can be used in case M 0 and m 6 0.
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4.20. Prove the mean value theorem.
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De ne F x f x f a x a f b f a . b a Then F a 0 and F b 0. Also, if f x satis es the conditions on continuity and di erentiability speci ed in Rolle s theorem, then F x satis es them also. Then applying Rolle s theorem to the function F x , we obtain F 0  f 0  f b f a 0; b a a<<b or f 0  f b f a ; b a a<<b
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4.21. Verify the mean value theorem for f x 2x2 7x 10, a 2, b 5.
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f 2 4, f 5 25, f 0  4 7. Then the mean value theorem states that 4 7 25 4 = 5 2 or  3:5. Since 2 <  < 5, the theorem is veri ed.
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4.22. If f 0 x 0 at all points of the interval a; b , prove that f x must be a constant in the interval.
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Let x1 < x2 be any two di erent points in a; b . By the mean value theorem for x1 <  < x2 , f x2 f x1 f 0  0 x2 x1 Thus, f x1 f x2 constant. From this it follows that if two functions have the same derivative at all points of a; b , the functions can only di er by a constant.
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4.23. If f 0 x > 0 at all points of the interval a; b , prove that f x is strictly increasing.
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Let x1 < x2 be any two di erent points in a; b . By the mean value theorem for x1 <  < x2 , f x2 f x1 f 0  > 0 x2 x1 Then f x2 > f x1 for x2 > x1 , and so f x is strictly increasing.
4.24. (a) Prove that
b a b a < tan 1 b tan 1 a < if a < b. 2 1 b 1 a2  3 4  1 < tan 1 < . (b) Show that 4 25 3 4 6
Since f 0 x 1= 1 x2 and f 0  1= 1 2 , we have by the mean value
(a) Let f x tan 1 x. theorem
DERIVATIVES
[CHAP. 4
tan 1 b tan 1 a 1 b a 1 2 Since  > a, 1= 1 2 < 1= 1 a2 .
a<<b Then
Since  < b, 1= 1 2 > 1= 1 b2 .
1 tan 1 b tan 1 a 1 < < 2 b a 1 b 1 a2 and the required result follows on multiplying by b a. (b) Let b 4=3 and a 1 in the result of part (a). Then since tan 1 1 =4, we have 3 4 1 < tan 1 tan 1 1 < 25 3 6 or  3 4  1 < tan 1 < 4 25 3 4 6
4.25. Prove Cauchy s generalized mean value theorem.
Consider G x f x f a fg x g a g, where is a constant. Then G x satis es the conditions of Rolle s theorem, provided f x and g x satisfy the continuity and di erentiability conditions of Rolle s f b f a theorem and if G a G b 0. Both latter conditions are satis ed if the constant . g b g a 0 Applying Rolle s theorem, G  0 for a <  < b, we have f 0  g 0  0 as required. or f 0  f b f a ; g 0  g b g a a<<b
L HOSPITAL S RULE 4.26. Prove L Hospital s rule for the case of the indeterminate forms (a) 0/0, (b) 1=1.
(a) We shall suppose that f x and g x are di erentiable in a < x < b and f x0 0, g x0 0, where a < x0 < b. By Cauchy s generalized mean value theorem (Problem 25), f x f x f x0 f 0  g x g x g x0 g 0  Then
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