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

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


3 Answers

jitender lakhanpal
62 Points
11 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

We are all IITians and here to help you in your IIT JEE preparation.

Now you can win by answering the questions on Discussion Forum. So help discuss any query on askiitians forum and become an Elite Expert League askiitian.

Now you score 5+15 POINTS by uploading your Pic and Downloading the Askiitians Toolbar : Click here to download the toolbar..  


manuj mittal
4 Points
11 years ago

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

Swapnil Saxena
102 Points
10 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)] 


Think You Can Provide A Better Answer ?

Provide a better Answer & Earn Cool Goodies See our forum point policy


Get your questions answered by the expert for free