Final answer:
To find the relative maxima and minima of the function f(x, y) = x² + xy + y² - 16y + 85, solve the system of equations formed by setting its partial derivatives to zero. Determine the nature of the critical points using the second derivative test and the Hessian matrix.
Step-by-step explanation:
To find the relative maximum and minimum values of the function f(x, y) = x² + xy + y² - 16y + 85, we first need to find the critical points where the gradient of f is zero. This involves calculating the partial derivatives of f with respect to x and y, setting them equal to zero, and solving for x and y.
The partial derivative with respect to x is 2x + y and with respect to y is x + 2y - 16. Setting them equal to zero gives us two equations:
2x + y = 0
x + 2y - 16 = 0
Solving this system of equations will give us the critical points. To determine whether these points are relative maxima, minima, or saddle points, we use the second derivative test. We find the second partial derivatives of f, f₁₁, f₂₂, and the mixed derivative f₁₂, and evaluate the determinant of the Hessian matrix at the critical points. If the determinant is positive and f₁₁ > 0, the point is a relative minimum; if the determinant is positive and f₁₁ < 0, it's a relative maximum; and if the determinant is negative, the point is a saddle point.