Advanced book on Mathematics Olympiad

134 3 Real Analysis

401.Prove that any convex polygonal surface can be divided by two perpendicular lines

into four regions of equal area.

402.Letf :I → Rbe a function defined on an interval. Show that iffhas the

intermediate value property and for anyy∈Rthe setf−^1 (y)is closed, thenfis

continuous.

403.Show that the function

fa(x)=

{

cos^1 x forx    = 0 ,

a forx= 0 ,

has the intermediate value property ifa∈[− 1 , 1 ]but is the derivative of a function

only ifa=0.

3.2.4 Derivatives and Their Applications..........................

A functionfdefined in an open interval containing the pointx 0 is called differentiable
atx 0 if

lim

h→ 0

f(x 0 +h)−f(x 0 )

h

exists. In this case, the limit is called the derivative offatx 0 and is denoted byf′(x 0 )

ordfdx(x 0 ). If the derivative is defined at every point of the domain off, thenfis simply

called differentiable.

The derivative is the instantaneous rate of change. Geometrically, it is the slope of

the tangent to the graph of the function. Because of this, where the derivative is positive

the function is increasing, where the derivative is negative the function is decreasing, and

on intervals where the derivative is zero the function is constant. Moreover, the maxima

and minima of a differentiable function show up at points where the derivative is zero,

the so-called critical points.

Let us present some applications of derivatives. We begin with an observation made

by F. Pop during the grading of USA Mathematical Olympiad 1997 about a student’s

solution. The student reduced one of the problems to a certain inequality, and the question

was whether this inequality is easy or difficult to prove. Here is the inequality and Pop’s

argument.

Example.Leta, b, cbe positive real numbers such thatabc=1. Prove that

a^2 +b^2 +c^2 ≤a^3 +b^3 +c^3.