Answer to Question #2357 in Calculus for Marcel

Question

Find the maximum and minimum values of the function f(x,y) = x2 ‐2y subject to the constraint
that x2 + y2 = 9 and please show calculations

Answer 1

We should use Lagrange multipliers and Extreme value theorem. Let's construct the system of equations:
2x = 2λ x;
-2 = 2λ y;
x² + y² = 9.
Hence: λ = 1, y = -1, x² = 8.
The function has one extreme value:
f(x,y) = 8 + 2 = 10.
f(√5, 2) = 5 - 2 = 3.
Thus f(x,y) = 10 is the maximum of the given function subject to constraint that x² + y² = 9.