Question

Find the local maximum and minimum values and saddle point(s) of the function. 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.)

f(x, y) = 3x³ − 9x + 9xy²

Answer 1

Pls, give thumbs up.

f = 3x^3 - 9x + 9xy^2

fx = partial der of f with x
fx = 9x^2 - 9 + 9y^2 = 0
x^2 + y^2 = 1

fy = 18xy = 0
x = 0 or y = 0

When x = 0, we get y = -1 or 1
When y = 0, we get x = -1 or 1

So, criticals are (0,-1) , (0,1) , (-1,0) and (1,0)

Now, fxx = 18x
fxy = 18y
fyy = 18x

Now, D = fxx*fyy - fxy^2

D = 324x^2 - 324y^2
and fxx = 18x

With (0,-1) ---> D < 0 ---> saddle
Wiht (0,1) ---> D < 0 ---> saddle
With (1,0) ---> D > 0 and fxx > 0 ---> minimum
With (-1,0) ---> D > 0 and fxx < 0 ---> maximum

Local max occurs at (-1,0) and value = f = 6
local min occurs at (1,0) and value = f = -6
saddle point = (0,-1) and (0,1)