Lagrange multipliers reveal maximums at (1,2) and (-1,-2), and minimums at (2,1) and (-2,-1) for the function xy subject to 4x² + y² = 8. The critical points arise from intersecting the function's gradient with the constraint's gradient scaled by a Lagrange multiplier.
Finding the extreme values of a function under a constraint requires balancing the function's direction of change with the constraint's restriction. Lagrange multipliers provide a powerful tool for this.
Here's how we apply it to the given problem:
- Form the Lagrange function: L(x, y, λ) = xy + λ(4x² + y² - 8)
- Find critical points: Take the partial derivatives of L with respect to x, y, and λ and set them equal to zero. This yields a system of three equations involving x, y, and λ.
- Solve the system: Solve the system of equations from step 2 to find the critical points (x, y) for potential extremum values.
- Classify critical points: Analyze the Hessian matrix of the Lagrange function evaluated at each critical point. A positive definite Hessian indicates a minimum, negative definite a maximum, and indefinite a saddle point.
In this case, we obtain four critical points: (1, 2), (-1, -2), (2, 1), and (-2, -1). The Hessian analysis reveals that (1, 2) and (-1, -2) are maximums, while (2, 1) and (-2, -1) are minimums.
Therefore, the function xy attains its maximum values of 2 at (1,2) and (-1,-2), and its minimum values of -2 at (2,1) and (-2,-1) subject to the constraint 4x² + y² = 8.