The purpose of using Lagrange Multiplier is to find the maximum or minimum value of a function subject to a constraint. In this case, we want to find the maximum or minimum value of the function x2−2xy+7y2 on the ellipse x2+4y2=1, while also satisfying the constraint of the ellipse equation.
To set up the Lagrange Multiplier equation, we first write out the function we want to optimize (x2−2xy+7y2) and the constraint equation (x2+4y2=1). Then, we introduce a new variable λ (the Lagrange Multiplier) and set up the following equation: ∇f(x,y) = λ∇g(x,y), where ∇f(x,y) is the gradient of the function and ∇g(x,y) is the gradient of the constraint equation.
To solve for the critical points, we first find the partial derivatives of the function and the constraint equation with respect to x and y. Then, we set those derivatives equal to λ times the partial derivatives of the function and constraint equation with respect to λ. We can then solve this system of equations to find the critical points.
To determine if the critical points are maximum, minimum, or neither, we use the second derivative test. We take the second derivative of the function and evaluate it at each critical point. If the second derivative is positive, then the critical point is a minimum. If the second derivative is negative, then the critical point is a maximum. If the second derivative is zero, then we need to use another method to determine the nature of the critical point.
The solution obtained from using Lagrange Multiplier is the maximum or minimum value of the function x2−2xy+7y2 on the ellipse x2+4y2=1, while also satisfying the constraint of the ellipse equation. This solution is significant because it allows us to optimize the function while taking into account the constraint, which may not be possible using other methods.