Question

Find the local maximum and minimum values and saddle point(s) of the function f(x, y) = x^3 - 75xy + 125y^3. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

a) Local max: DNE, Local min: DNE, Saddle point: (0,0,0)
b) Local max: (0,0,0), Local min: DNE, Saddle point: DNE
c) Local max: (0,0,0), Local min: (0,0,0), Saddle point: DNE
d) Local max: DNE, Local min: (0,0,0), Saddle point: DNE

Answer 1

To determine the nature of the critical points for the function f(x, y) = x^3 - 75xy + 125y^3, the first and second partial derivatives were computed, leading to the conclusion that (0,0) is a saddle point with no local maxima or minima.

Step-by-step explanation:

To find the local maximum, local minimum, and saddle points of the function f(x, y) = x^3 - 75xy + 125y^3, we need to calculate the first and second partial derivatives:

• fx(x, y) = 3x^2 - 75y
• fy(x, y) = -75x + 375y^2
• f__xx(x, y) = 6x
• fyy(x, y) = 750y
• fxy(x, y) = fyx(x, y) = -75

We set the first partial derivatives to zero to find critical points:

• 3x^2 - 75y = 0
• -75x + 375y^2 = 0

From these equations, we get the critical point at (0, 0). We then use the second derivative test to determine the nature of this critical point. The determinant of the Hessian matrix H at (0,0) is:

H = f__xxfyy - (fxy)^2 = (6x)(750y) - (-75)^2

At the critical point (0, 0), the determinant of Hessian matrix H is:

H(0,0) = (6(0))(750(0)) - (-75)^2 = -5625

Since H(0,0) < 0, the point (0,0) is a saddle point. There is no local maximum or local minimum.

The correct answer is option a) Local max: DNE, Local min: DNE, Saddle point: (0,0,0).