9.11. EXERCISES 219

23. Let p,q> 1 and satisfy 1p +

1q = 1. Consider the function x= t p−1. Then solving for t,

you get t = x1/(p−1) = xq−1. Explain this. Now let a,b≥ 0. Sketch a picture to showwhy

∫ b0 xq−1dx+

∫ a0 t p−1dt ≥ ab. Now do the integrals to obtain a very important

inequality bq

q + ap

p ≥ ab. When will equality hold in this inequality?

24. Suppose f ,g are two Riemann Stieltjes integrable functions on [a,b] with respect toF, an increasing function. Verify Holder’s inequality.

∫ b

a| f | |g|dF ≤

(∫ b

a| f |p dF

)1/p(∫ b

a|g|q dF

)1/q

,1p+

1q= 1, p > 1

Hint: Do the following. Let

A =

(∫ b

a| f |p dF

)1/p

,B =

(∫ b

a|g|q dF

)1/q

.

Then let a = | f |A ,b = |g|

B and use the wonderful inequality of Problem 23.

25. Let F (x) =∫ x3

x2t5+7

t7+87t6+1 dt. Find F ′ (x) .

26. Let F (x) =∫ x

21

1+t4 dt. Sketch a graph of F and explain why it looks the way it does.

27. Let a and b be positive numbers and consider the function F (x) =∫ ax

01

a2+t2 dt +∫ a/xb

1a2+t2 dt. Show that F is a constant.

28. Solve the following initial value problem from ordinary differential equations whichis to find a function y such that

y′ (x) =x4 +2x3 +4x2 +3x+2

x3 + x2 + x+1, y(0) = 2.

29. If F,G ∈∫

f (x) dx for all x ∈ R, show F (x) = G(x)+C for some constant, C. Usethis to give a different proof of the fundamental theorem of calculus which has forits conclusion

∫ ba f (t)dt = G(b)−G(a) where G′ (x) = f (x) .

30. Suppose f is continuous on [a,b]. Show there exists c ∈ (a,b) such that

f (c) =1

b−a

∫ b

af (x) dx.

Hint: You might consider the function F (x) ≡∫ x

a f (t) dt and use the mean valuetheorem for derivatives and the fundamental theorem of calculus. In a sense, thiswas done in the chapter, but this one is more specific and note that here c ∈ (a,b),the open interval.

31. Use the mean value theorem for integrals, Theorem 9.1.5 or the above problem toconclude that

∫ a+1a ln(t)dt = ln(x)≤ ln

(a+ 1

2

)for some x ∈ (a,a+1). Hint: Con-

sider the shape of the graph of ln(x) in the following picture. Explain why if x is

9.11.23.24.25.26.27,28.29.30.31.EXERCISES 219Let p,q > 1 and satisfy D + j = 1. Consider the function x = t?~!. Then solving for f,you get t = x!/(P-)) = x4-!, Explain this. Now let a,b > 0. Sketch a picture to showwhy So xd! dx + Jot? !dt > ab. Now do the integrals to obtain a very importantinequality n + “ > ab. When will equality hold in this inequality?Suppose f,g are two Riemann Stieltjes integrable functions on [a,b] with respect toF, an increasing function. Verify Holder’s inequality.b b '/P / pb Vay 4[intiar<(['rrar) (fear) 24 =1,.p>1Hint: Do the following. Letb 1/p b 1/q(/ "ar ) w= (J ear ) |lfb= lel and use the wonderful inequality of Problem 23.AThen let a=3Let F (x) = fb StL dt. Find F'(x).Let F (x) = fr a dt. Sketch a graph of F and explain why it looks the way it does.Let a and b be positive numbers and consider the function F (x) = {>* apdt +ser “ pee dt. Show that F is a constant.Solve the following initial value problem from ordinary differential equations whichis to find a function y such that_ x4 42x93 + Ax? +3442/y@) B+x2+x4+1If F,G € f f (x) dx for all x € R, show F (x) = G(x) +C for some constant, C. Usethis to give a different proof of the fundamental theorem of calculus which has forits conclusion ? f (t) dt = G(b) — G(a) where G! (x) = f (x).Suppose f is continuous on [a,b]. Show there exists c € (a,b) such that1 bsl f (x) dx.Hint: You might consider the function F (x) = J? f(t) dt and use the mean valuetheorem for derivatives and the fundamental theorem of calculus. In a sense, thiswas done in the chapter, but this one is more specific and note that here c € (a,b),the open interval.fle)=Use the mean value theorem for integrals, Theorem 9.1.5 or the above problem toconclude that [@*! In(t)dt = In(x) < In (a+ 4) for some x € (a,a+1). Hint: Con-sider the shape of the graph of In(x) in the following picture. Explain why if x is