It is obviously greater than negative infinity!

Algebra Level 4

For all real x,y the value of x 2 + 2 x y + 3 y 2 6 x 2 y x^{2} +2xy +3y^{2} -6x-2y cannot be less than k -k . k is


The answer is 11.

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

3 solutions

Kenny Lau
Sep 8, 2014

(Waiting for a non-calculus approach)

  • Let z = x 2 + 2 x y + 3 y 2 6 x 2 y z=x^2+2xy+3y^2-6x-2y .
  • At the minimum of the 3D graph, the slope of tangent plane is 0.
  • Therefore, the slope of tangent line parallel to the x-axis is 0.
  • Same goes with the line parallel to the y-axis.
  • d z d x = 2 x + 2 y 6 = 0 \frac{dz}{dx}=2x+2y-6=0
  • d z d y = 2 x + 6 y 2 = 0 \frac{dz}{dy}=2x+6y-2=0
  • Solve equation to give x=4 and y=-1
  • Substitute the values to z to give -11.
Joel Tan
Sep 11, 2014

Suppose that x 2 + 2 x y + 3 y 2 6 x 2 y = ( x + y 3 ) 2 + 2 y 2 + 4 y 9 = k x^{2}+2xy+3y^{2}-6x-2y=(x+y-3)^{2}+2y^{2}+4y-9=-k has a solution. Then ( x + y 3 ) 2 = 2 y 2 4 y + ( 9 k ) 0 (x+y-3)^{2}=-2y^{2}-4y+(9-k) \geq 0 hence 2 y 2 + 4 y + ( k 9 ) = 2 ( y + 1 ) 2 + ( k 11 ) 0 2y^{2}+4y+(k-9)=2 (y+1)^{2}+(k-11) \leq 0 Thus k k is at most 11 11 . Equality can be achieved when y = 1 , x = 4 y=-1, x=4 . The answer follows.

Arturo Presa
May 15, 2017

It is easy to see that x 2 + 2 x y + 3 y 2 6 x 2 y = ( x + y 3 ) 2 + 2 ( y + 1 ) 2 11. x^{2} +2xy +3y^{2} -6x-2y =(x+y-3)^2+2(y+1)^2-11. Since ( x + y 3 ) 2 + 2 ( y + 1 ) 2 0 (x+y-3)^2+2(y+1)^2\geq 0 , then x 2 + 2 x y + 3 y 2 6 x 2 y = ( x + y 3 ) 2 + 2 ( y + 1 ) 2 11 11. x^{2} +2xy +3y^{2} -6x-2y =(x+y-3)^2+2(y+1)^2-11\geq -11. Additionally the equality takes place if x + y 3 = 0 x+y-3=0 and y + 1 = 0 , y+1=0, that is, when x = 4 , x=4, and y = 1. y=-1. Therefore, the answer to the question is 11 . \boxed{11}.

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...