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

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) = x2 + xy + y2 + 8y

And th local maximum and minimum value5 and sad p in a the unctinn. If nu have th dr an inna g hin Software, aph tha un on w th a dom n and w p int hatras ala ha! ortant asp cts ot a un tion. Enter nu an w ars a enter DME.) /(x, y)=x2 + xy + 2 + ey local minimum value(s) saddle point(s) ex, y,n

Explanation / Answer

f = x^2 + xy + y^2 + 8y

fx = partial der of f woth x
fx = 2x + y = 0

fy = x + 2y + 8 = 0

Solving :
2x + y = 0
x + 2y = -4

We get :
x = 4/3 and y = -8/3

So, this is the only critical.....

fxx = double der with x
fxx = 2

fyy = 2

fxy = 1

So, discriminant, D = fxx*fyy - fxy^2
D = 2*2 - 1^2
D = 4 - 1
D = 3

So, we have D and fxx values > 0

So, the critical we got was a MINIMUM

(4/3 , -8/3)

Now, we find function value also....

f = x^2 + xy + y^2 + 8y

f = (4/3)^2 + (4/3)(-8/3) + (-8/3)^2 + 8(-8/3)

f = -16

So, answers :

Local max : DNE
Local min : (4/3 , -8/3 , -16)
Saddle : DNE

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