# how to find range of x^2 + 2xy + 3y^2 -6x -2y .dont use cauchy schwartz inequality.

jitender lakhanpal
62 Points
12 years ago

Dear Manuj,

make the expression quadratic in x taking y as constant

 x^2 + 2xy + 3y^2 -6x -2y = 0

and that will be

 x^2 +(2y-6)x+(3y2 -2y) = 0

now the range can be found out by taking discriminant greater than equal to 0

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jitender

manuj mittal
4 Points
12 years ago

thanks for ur efforts but here it is not given tht equation has real roots.it may be possible equation is >0 for all real x and y.so how can we use discriminant >=0???.....plzz ans...

Swapnil Saxena
102 Points
12 years ago

Hi Manuj,

Differentiate the equation though the method of differentiation of implicit functions and put the dy/dx = 0 (at maxima and minima, slope of curve=0 ).

it will give u a equation of form 2x+2y-6=0 ==> x+y= 3 ==> x=3-y

Now  substitute x by (3-y) in the equation x2 + 2xy + 3y2 -6x -2y = 0  .

It will give you a quardatic equation in y. Find the roots of the equation and they will be the respective maximas and minimas of the equation.

So the range of this function will be [1/2(-2-root(22)),1/2(root(22)-2)]