Mathematical Proofs

WMA12/01 IAL (Edexcel) P2/Oct/2021/Q9 (Proof by Deduction and Counter Example)

• Prove that for all positive values of x and y,​​ 

x+y2≥xy 

(3)​​ 

• Prove by counter-example that this inequality does not hold when x and y are both negative.​​ 

(1)

SOLUTION

a-​​ 

First, we are going to use jotting method to find a good starting point, but remember, jotting is not the part of the proof.​​ 

Jotting

x+y2≥xy 

x+y22≥xy2

x+y24≥xy

x2+2xy+y2≥4xy

x2-2xy+y2≥0

x-y2≥0

Proof​​ 

We always start a proof from a known or settled fact like in this case, we know that the expression squared is always​​ ≥​​ 0.​​ 

Consider,​​ For all real x values 

x-y2≥0 

x2-2xy+y2≥0

x2-2xy+y2≥0

The statement we have to proof has​​ x – y2​​ on the left side of the inequality therefore, we will be using the completing square method to solve this inequality. ​​ 

x2-2xy+y2+4xy≥4xy

x2+2xy+y2≥4xy

x+y2≥4xy

Taking square root on both sides.

x+y2≥4xy

Since​​ x>0​​ and​​ y>0,​​ x+y2​​ is equal to only positive x+y​​ 

x+y=2xy

x+y2=xy

Hence, proved.

b-​​ 

For​​ x=-1​​ and​​ y=-1, on putting in​​ 

x+y2=xy

(-1)+(-1)2=(-1)(-1)

-22=1

-1≠1

Hence, the statement doesn’t hold when x and y are both negatives.​​