Final answer:
By computing the first and second partial derivatives of the function and applying the second derivative test, it is determined that the function has a saddle point at (0, 0, 0) and no local maximum or minimum.
Step-by-step explanation:
To find the local maximum, minimum values, and saddle points for the function f(x,y)=x^3−75xy+125y^3, we start by finding the first partial derivatives of the function with respect to both x and y, and then solve for points where these derivatives are zero to find critical points. Next, we examine the second partial derivatives and use the second derivative test by computing the Hessian determinant at the critical points to determine the nature of these points.
The first derivatives are:
• f_x = 3x^2 - 75y (partial derivative with respect to x)
• f_y = -75x + 375y^2 (partial derivative with respect to y)
Solving the system of equations f_x = 0 and f_y = 0 gives us the critical point (0, 0).
The second derivatives are:
• f_xx = 6x
• f_yy = 750y
• f_xy = f_yx = -75
The Hessian determinant, D, at the critical point (0, 0) is D = f_xx * f_yy - (f_xy)^2 = (6x)(750y) - (-75)^2. Plugging the critical point into this determinant, we get D = (6(0))(750(0)) - (-75)^2 = -5625, which is less than zero, indicating a saddle point at (0, 0).
Since the Hessian determinant is negative, we have a saddle point and no local maximum or minimum. Thus, the correct answer is:
Local Maximum: DNE
Local Minimum: DNE
Saddle Point: (0, 0, 0)