Question

# Find the local maximum and minimum values and saddle point(s) of the function. If you have...

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

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)

#### Earn Coins

Coins can be redeemed for fabulous gifts.