Optimization inequality constraint

[oswald]

Homework Statement

Minimize 2x²+2y²-2xy-9y subject
4x + 3y =,< 10 ,
y - 4x² =,< -2
x >,= 0
and y >,= 0.


I don't undersant this:

"This equation has no nonnegative root, which contradicts a nonnegativity
constraint."
and how we solve
-16x² + 2x + 17 + h2 = 0

 

[HallsofIvy]

Homework Statement

Minimize 2x²+2y²-2xy-9y subject
4x + 3y =,< 10 ,
y - 4x² =,< -2
x >,= 0
and y >,= 0.


I don't undersant this:

"This equation has no nonnegative root, which contradicts a nonnegativity
constraint."

and how we solve
-16x² + 2x + 17 + h2 = 0

Do you know how to complete the square? That would seem to me to be the best way to solve that equation. (The "h2" here is supposed to be \lambda_2, one of the Lagrange multipliers.)

 

[oswald]

complete the square on -16x² + 2x + 17 + h2 = 0 or -16x² + 2x + 17 = 0, because i found positive and negative root [ x=~+-1] on -16x² + 2x + 17=0 and i don't know how solve with λ2.

 

[HallsofIvy]

Okay, since you are basically saying you do NOT know how to complete the square,
-16x^2+ 2x+ 17= -16(x^2- (1/8)x)= -16(x^2- (1/8)x+ (1/256)- 256)+ 17= -16(x- 1/16)^2+ 17+ 1/16.
So ]-16x^2+ 2x+ 17+ \lambda_2 can only be equal to 0 if \lambda+ 17+ 1/16&gt;0 or \lambda&lt; -(17+ 1/16). That's what violates the "nonegativity constraint", that \lambda can't be negative.

 

[oswald]

why
λ + 17 + 1/16 > 0?