\hspace{-16}$Find Max. and Min. value of $\mathbf{f(x,y)=\frac{2x^2+7y^2-12xy}{x^2+y^2}}$\\\\ Where $\mathbf{x,y}$ are Real no. not Simultaneously Zero
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5 Answers
71Yes It is.. Now It's verified.
See the posts here http://www.artofproblemsolving.com/Forum/viewtopic.php?f=36&t=447496
62\frac{4x^2+9y^2-12xy}{x^2+y^2}-2=\frac{(2x-3y)^2}{x^2+y^2} -2
Now take either y/x as t
or x=r sin theta, y = r cos theta...
and then it is a simple maxima minima problem?
71Sir, i got it somehow using partial derivative.. partial differentiating it w.r.t x or y, it will give two values of x (w.r.t y). Putting that in above we get y2 or x2 terms get cancelled giving perfect -2 and 11 as answers?
Secondly I had used what you said as first method.. (y/x = t)
Now, As you have shown one arrangement , the left side is always positive so min. becomes -2. Similar could be done for upper bound..(in the AoPs link mentioned )
