Final answer:
The absolute minimum value of the function f(x, y) = 2x^3 + y^4 on the region defined by x^2 + y^2 ≤ 100 is 0, which occurs at the point (0, 0) after evaluating the interior and boundary of the circle.
Step-by-step explanation:
The student has been given a multivariable function f(x, y) = 2x3 + y4 and is asked to find the absolute minimum value on the region where x2 + y2 ≤ 100. This region is a circle with radius 10 centered at the origin in the xy-plane. To find the absolute minimum, we need to check both the interior and the boundary of the region.
First, we examine the interior by finding the critical points where the partial derivatives with respect to x and y both equal zero. Calculating the partial derivatives, we get:
- ∂f/∂x = 6x2 and ∂f/∂y = 4y3
Setting these equal to zero implies that x = 0 and y = 0 are critical points. Substituting these into the function, we find f(0, 0) = 0 which is a possible minimum. Now we examine the boundary, which is when x2 + y2 = 100. Here, we can use the method of Lagrange multipliers or simply recognize that on the boundary, the x and y terms will only add positive values to the function, suggesting the minimum occurs at the origin. Therefore, the absolute minimum value of the function in the given region is 0, occurring at the point (0, 0).