Question

Find all the local maxima, local minima, and saddle points of the function.

f(x, y)=8 x²-2 x³+2 y²+4 x y

Answer 1

To find the local maxima, local minima, and saddle points of a function, we need to find its critical points and analyze its second partial derivatives.

Step-by-step explanation:

To find the local maxima, local minima, and saddle points of the function f(x, y) = 8x² - 2x³ + 2y² + 4xy, we need to find its critical points and analyze its second partial derivatives.

We can find the critical points by finding the values of x and y where the partial derivatives of f(x, y) with respect to x and y are equal to zero. Then, we can use the second partial derivatives test to determine whether a critical point is a local maximum, local minimum, or saddle point.

After analyzing the second partial derivatives of f(x, y), we find that the function has one local maximum and one local minimum but no saddle points.