Given that \(\displaystyle{2}{x}^{{2}}-{4}{x}{y}+{6}{y}^{{2}}={9}\), find the biggest and

Landen Barber

Landen Barber

Answered question

2022-03-27

Given that 2x24xy+6y2=9, find the biggest and lowest value of 2xy

Answer & Explanation

Jermaine Lam

Jermaine Lam

Beginner2022-03-28Added 11 answers

Step 1
2x24xy+6y2=9
and let
k=2xyy=2xk
2x24x(2xk)+6(2xk)2=9
2x28x2+4xk+6(4x2+k24xk)=9
6x2+4xk+24x2+6k224xk9=0
18x220kx+6k29=0
For real roots, Discriminant 0
400k2418(6k29)0
100k2108k2+1620
8k2+1620k28140
92k92
Kamora Campbell

Kamora Campbell

Beginner2022-03-29Added 13 answers

Step 1
The trick is to notice that:
4(2xy)2=9(2x24xy+6y2)2(x5y)29(2x24xy+6y2)=81
It follows that
922xy92
The minimum is attained when (x,y)=(d52,d12)
and the maximum when (x,y)=(d52,d12)

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