Question

Find All The Local Maxima, Local Minima, And Saddle Points Of The Function.

F(X,Y)=2x²+3xy+4y²−7x−11y
A. A local maximum occurs at (Type an ordered pair. Use a comma to separate answers as needed.) The local maximum value(s) is/are (Type an exact answer. Use a comma to separate answers as needed.)
B. There are no local maxima.

Answer 1

By solving the system of equations from the first partial derivatives of the function f(x, y), the critical point at (1, 1) is found, and using the second derivative test, it is determined to be a local minimum. There are no local maxima or saddle points for this function.

Step-by-step explanation:

To find the local maxima, local minima, and saddle points of the function f(x,y) = 2x² + 3xy + 4y² - 7x - 11y, we first need to calculate the partial derivatives of f with respect to x and y.

The partial derivative with respect to x is fx(x,y) = 4x + 3y - 7, and the partial derivative with respect to y is fy(x,y) = 3x + 8y - 11. To find critical points, we set these partial derivatives equal to zero and solve the resulting system of equations.

Solving the system:

• 4x + 3y = 7
• 3x + 8y = 11

we find the critical point (1, 1). Next, we use the second derivative test to determine the nature of the critical point. We calculate the second partial derivatives:

• f(x,y) = 4
• f(x,y) = 8
• f(x,y) = fyx(x,y) = 3

Then we compute the Hessian determinant at the critical point H = fx * fy - (fxy)^2. With the values for the second partial derivatives, we have H = 4 * 8 - 3 * 3 = 32 - 9 = 23.

Since H > 0 and fx > 0 at the critical point, we conclude that f(x,y) has a local minimum at (1, 1).

There are no local maxima for the given function, and since the determinant is positive and we found a local minimum, there are no saddle points.