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*(

Answer #1

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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)**

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

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

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
local maximum value(s)
local minimum value(s)
saddle point(s)

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) =
x3 + y3 −
3x2 − 6y2
− 9x

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) = x3 + y3 − 3x2 − 3y2 − 9x

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) =
x3 + y3 −
3x2 − 3y2
− 9x

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) =
x3 + y3 −
3x2 − 6y2
− 9x

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) =
x3 + y3 −
3x2 − 6y2
− 9x

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) =
x3 + y3 −
3x2 − 9y2
− 9x
local maximum value(s)
????
local minimum value(s)
?????
saddle point(s)
(x, y) =????

1) 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) =
x3 + y3 −
3x2 − 9y2
− 9x

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