# Optimization inequality constraint (1 Viewer)

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#### oswald

1. The problem statement, all variables and given/known data

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

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#### HallsofIvy

1. The problem statement, all variables and given/known data

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.)

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#### 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 dont 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 $\lambda+ 17+ 1/16>0$ or $\lambda< -(17+ 1/16)$. That's what violates the "nonegativity constraint", that $\lambda$ can't be negative.

#### oswald

why
λ + 17 + 1/16 > 0?

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