Question

Determine the global extreme values of the function f(x,y) = x³ + x²y + 7y²?

Answer 1

To find the global extreme values of the function f(x,y) = x³ + x²y + 7y², determine where the gradient is zero and use the second derivative test or function behavior analysis to classify these points and identify the global maximum and minimum values.

Step-by-step explanation:

To determine the global extreme values of the function f(x,y) = x³ + x²y + 7y², we need to find the points where the gradient of f is zero and then classify these points using the second derivative test or by analyzing the behavior of the function. To do this, we first find the partial derivatives of f with respect to x and y.

1. Calculate the partial derivative with respect to x: fx(x,y) = 3x² + 2xy.
2. Calculate the partial derivative with respect to y: fy(x,y) = x² + 14y.
3. Set these partial derivatives equal to zero and solve the system of equations to find critical points. This involves solving 3x² + 2xy = 0 and x² + 14y = 0.
4. Analyze the second partial derivatives and apply the second derivative test to classify the nature of the critical points.
5. Inspect the behavior of f at the boundaries or at infinity to ensure that the extreme values are global.

After identifying the candidates for the global extreme values, compare the function values at these points to find the highest and lowest values of f, which would be the global maximum and minimum, respectively.