mathematics for economists

Question

Find the extreme values of f(x,y) = 6y²+x² subject to the constraint x²+y²= 4.

Answer 1

You don't have to use Calculus for this.

f(x,y) = 6y² + x² may be rewritten 5y² + x² + y² = 5y² + 4

If x² + y² = 4, as both x² + y² are non-negative, the minimum value of y² is 0, when x = ±2 and y = 0, and the maximum value of y² is 4, when y = ±2 and x = 0

The maximum value of 6y² + x² = 5y² + 4 occurs when y = ±2 and x = 0 and is 5*4 + 4 = 24

The minimum value of 6y² + x² = 5y² + 4 occurs when x = ±2 and y = 0 and is 5*0 + 4 = 4