How do you minimize and maximize #f(x,y)=x^3-y^2-xy# constrained to #xy=4#?
Substitute
By signing up, you agree to our Terms of Service and Privacy Policy
To minimize and maximize the function ( f(x,y) = x^3 - y^2 - xy ) subject to the constraint ( xy = 4 ), we can use the method of Lagrange multipliers. The Lagrangian is given by:
[ L(x, y, \lambda) = x^3 - y^2 - xy + \lambda(xy - 4) ]
Where ( \lambda ) is the Lagrange multiplier. We need to find critical points by setting the partial derivatives of ( L ) with respect to ( x ), ( y ), and ( \lambda ) equal to zero:
- ( \frac{\partial L}{\partial x} = 3x^2 - y - \lambda y = 0 )
- ( \frac{\partial L}{\partial y} = -2y - x - \lambda x = 0 )
- ( \frac{\partial L}{\partial \lambda} = xy - 4 = 0 )
From equation 3, we have ( xy = 4 ). Substituting this into equations 1 and 2, we get:
- ( 3x^2 - y - 4\lambda = 0 ) (equation 4)
- ( -2y - x - 4\lambda = 0 ) (equation 5)
Multiplying equation 4 by 2 and adding it to equation 5, we get:
[ -6x^2 - 2x = 0 ]
This simplifies to ( x(x+1) = 0 ), so ( x = 0 ) or ( x = -1 ). If ( x = 0 ), then ( y = \pm 2 ). If ( x = -1 ), then ( y = \mp 4 ).
Thus, the critical points are ( (0, 2) ), ( (0, -2) ), ( (-1, -4) ), and ( (-1, 4) ). To determine which of these points correspond to the minimum or maximum of ( f ), we evaluate ( f ) at each point:
- ( f(0, 2) = 0^3 - 2^2 - 0\cdot2 = -4 )
- ( f(0, -2) = 0^3 - (-2)^2 - 0\cdot(-2) = -4 )
- ( f(-1, -4) = (-1)^3 - (-4)^2 - (-1)\cdot(-4) = 3 + 16 + 4 = 23 )
- ( f(-1, 4) = (-1)^3 - 4^2 - (-1)\cdot4 = -1 - 16 + 4 = -13 )
Therefore, the minimum value of ( f ) is -4 and occurs at ( (0, 2) ) and ( (0, -2) ), while the maximum value is 23 and occurs at ( (-1, -4) ).
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
- Two sides of a triangle are 6 m and 7 m in length and the angle between them is increasing at a rate of 0.07 rad/s. How do you find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length is pi/3?
- How do you use Newton's method to find the approximate solution to the equation #e^x+x=4#?
- An open top box is to be constructed so that its base is twice as long as it is wide. Its volume is to be 2400cm cubed. How do you find the dimensions that will minimize the amount of cardboard required?
- How do you use linear Approximation to find the value of #(1.01)^10#?
- How do you find the linearization at a=3 of # f(x) = 2x³ + 4x² + 6#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7