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Question
If f(x,y)=−10x2+6xy−6x−y2−1 where x,y∈R is a quadratic polynomial in two variable, then which of the following options is/are true?
If f(x,y)=−10x2+6xy−6x−y2−1 where x,y∈R is a quadratic polynomial in two variable, then which of the following options is/are true?
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Solution
The correct option is C The maximum value of f(x,y) is occurs at x=−3 and y=−9
f(x,y)=−10x2+6xy−6x−y2−1=−(9x2−6xy+y2)−(x2+6x+9)+8=−(3x−y)2−(x+3)2+8⇒f(x,y)=8−[(3x−y)2+(x+3)2]
As
(3x−y)2+(x+3)2≥0⇒8−[(3x−y)2+(x+3)2]≤8∴f(x,y)≤8
The maximum value of f(x,y)=8, when
x+3=0, 3x−y=0⇒x=−3,y=−9
f(x,y)=−10x2+6xy−6x−y2−1=−(9x2−6xy+y2)−(x2+6x+9)+8=−(3x−y)2−(x+3)2+8⇒f(x,y)=8−[(3x−y)2+(x+3)2]
As
(3x−y)2+(x+3)2≥0⇒8−[(3x−y)2+(x+3)2]≤8∴f(x,y)≤8
The maximum value of f(x,y)=8, when
x+3=0, 3x−y=0⇒x=−3,y=−9
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