Find the extreme values of the function subject to the given constraint.
-$$f ( x , y ) = x ^ { 2 } + 4 y ^ { 3 } , x ^ { 2 } + 2 y ^ { 2 } = 2$$
A) Maximum: 8 at (2, 1); minimum: -31 at (1, -2)
B) Maximum: 8 at (2, 1); minimum: -4 at (0, -1)
C) Maximum: 4 at (0, 1); minimum: -4 at (0, -1)
D) Maximum: 4 at (0, 1); minimum: -31 at (1, -2)

Question

Quizplus · Accepted Answer

To find the extreme values of $f(x, y) = x^2 + 4y^3$ subject to the constraint $x^2 + 2y^2 = 2$, we can use the method of Lagrange multipliers or directly substitute the constraint into the function to eliminate one variable. The constraint can be rearranged to express $x^2$ in terms of $y^2$, and then substituted into $f(x, y)$ to find the extreme values in terms of a single variable. After finding the critical points, we can determine that the maximum value of the function, given the constraint, is 4 at (0, 1), and the minimum value is -4 at (0, -1).

Find the Extreme Values of the Function Subject to the Given