Advanced book on Mathematics Olympiad

3.2 Continuity, Derivatives, and Integrals 159

480.Letf:[ 0 ,∞)→[ 0 ,∞)be a continuous, strictly increasing function withf( 0 )=

0. Prove that
   ∫a

0

f(x)dx+

∫b

0

f−^1 (x)dx≥ab

for all positive numbersaandb, with equality if and only ifb=f(a). Heref−^1

denotes the inverse of the functionf.

481.Prove that for any positive real numbersx, yand any positive integersm, n,

(n− 1 )(m− 1 )(xm+n+ym+n)+(m+n− 1 )(xmyn+xnym)

≥mn(xm+n−^1 y+ym+n−^1 x).

482.Letfbe a nonincreasing function on the interval[ 0 , 1 ]. Prove that for anyα∈
( 0 , 1 ),

α

∫ 1

0

f(x)dx≤

∫α

0

f(x)dx.

483.Letf:[ 0 , 1 ]→[ 0 ,∞)be a differentiable function with decreasing first derivative,
and such thatf( 0 )=0 andf′( 0 )>0. Prove that
∫ 1

0

dx

f^2 (x)+ 1

≤

f( 1 )

f′( 1 )

Can equality hold?

484.Prove that any continuously differentiable functionf :[a, b]→Rfor which
f(a)=0 satisfies the inequality
∫b

a

f(x)^2 dx≤(b−a)^2

∫b

a

f′(x)^2 dx.

485.Letf(x)be a continuous real-valued function defined on the interval[ 0 , 1 ].
Show that
∫ 1

0

∫ 1

0

|f(x)+f(y)|dxdy≥

∫ 1

0

|f(x)|dx.

3.2.11 Taylor and Fourier Series..................................

Some functions, called analytic, can be expanded around each point of their domain in a
Taylor series