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USE MONOTONY
Graphical representation
SucCess!
SOLVING ECUATIONS BY GRAPHICAL METHOD
USE CONVEXITY
Definition:
A function f:I->R is convex if :
for any x1,x2 of I , for any tE[0,1] ,
the following inequality holds
f(tx1+(1-t) x2 )≤tf(x1 )+(1-t)f(x2 )

Function is strictly convex for X1 ≠ X2 and inequality is strict
PROPERTY
If f:I->R is strictly convex and g:I->R is concave equation f(x)=g(x) has at most two solutions.
PROPERTY

If f:I->R is strictly convex on the interval I, and g:I->R is a linear function and then the equation f(x)=g(x) has at most two solutions in the interval I.
PROPERTY
Let ecuation f(x)=c where c is a constant
and f is strictly monotone then there is a
unique solution to the equation.
PROPERTY
If functions f and g are strictly monotone on the interval I, the monotony different, at least one being strictly monotone, then the equation f(x)= g (x) has at most one solution on the interval I.
Demonstration:
Let f be strictly increasing, g decreasing on I. Assume that there are at least two different solutions x1, x2 in the interval I of the equation.
let x1 < x2. Then f(x1 )=g(x1 )≥ g(x2) =f(x2).
Contradiction with f strictly increasing on I.
EXAMPLE
Solve the equation
5^X=4X+1

Solution:
Check x = 0 and x = 1 are solutions. Since f,g: R-> R f(x)=5 is strictly convex and g (x) = 4x + 1 is linear function . Ecuation f(X)=g(x) has at most two real solutions.

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self-evaluation

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Thanks!