Final answer:
The extreme values of the function f(x, y) = 9x²+5y² given the constraint x²+y² = 1 are a minimum of 5 at (0, ±1) and a maximum of 9 at (±1, 0).
Step-by-step explanation:
To find the extreme values of the function f(x, y) = 9x² + 5y² subject to the constraint x² + y² = 1, we can use the method of Lagrange multipliers or simply substitute the constraint into the function. By substituting y² as 1 - x² into the function, we get f(x, y) = 9x² + 5(1 - x²) = 4x² + 5. The extreme values will occur when x = ±1 and y = 0 for the maxima, and when x = 0 and y = ±1 for the minima.
To find the maximum value, we substitute x = ±1 into f(x, y), which gives us f(±1, 0) = 9. To find the minimum value, we substitute y = ±1, which gives us f(0, ±1) = 5.
Therefore, the correct option would be 'A) Minimum: 5 at (0, ±1); maximum: 9 at (±1,0)'.