Advanced book on Mathematics Olympiad

3.2 Continuity, Derivatives, and Integrals 137

404.For a nonzero real numberxprove thatex>x+1.

405.Find all positive real solutions to the equation 2x=x^2.

406.Letf:R→Rbe given byf(x)=(x−a 1 )(x−a 2 )+(x−a 2 )(x−a 3 )+(x−
a 3 )(x−a 1 )witha 1 ,a 2 ,a 3 real numbers. Prove thatf(x)≥0 for all real numbers
xif and only ifa 1 =a 2 =a 3.

407.Determine

max

z∈C,|z|= 1

|z^3 −z+ 2 |.

408.Find the minimum of the functionf:R→R,

f(x)=

(x^2 −x+ 1 )^3

x^6 −x^3 + 1

.

409.How many real solutions does the equation

sin(sin(sin(sin(sinx))))=

x

3

have?

410.Letf:R→Rbe a continuous function. Forx∈Rwe define

g(x)=f(x)

∫x

0

f(t)dt.

Show that ifgis a nonincreasing function, thenfis identically equal to zero.

411. Letfbe a function having a continuous derivative on[ 0 , 1 ]and with the property
     that 0<f′(x)≤1. Also, suppose thatf( 0 )=0. Prove that
     [∫ 1

0

f(x)dx

] 2

≥

∫ 1

0

[f(x)]^3 dx.

Give an example in which equality occurs.

412.Letx, y, zbe nonnegative real numbers. Prove that
(a)(x+y+z)x+y+zxxyyzz≤(x+y)x+y(y+z)y+z(z+x)z+x.
(b)(x+y+z)(x+y+z)
2
xx
2
yy
2
zz
2
≥(x+y)(x+y)
2
(y+z)(y+z)
2
(z+x)(z+x)
2
.

Derivatives have an important application to the computation of limits.

L’Hôpital’s rule.For an open intervalI, if the functionsfandgare differentiable on
I{x 0 },g′(x) = 0 forx∈I,x =x 0 , and eitherlimx→x 0 f(x)=limx→x 0 g(x)= 0