What is the extreme f(x,y)=xln(x+y) ?

The extreme f(x,y)=xln(x+y) is Saddle(0,1)

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extreme f(x,y)=xln(x+y)

extreme points

f (x, y) = x ln (x + y)

Saddle (0, 1)

Solution steps

Find the critical points:

(0, 1)

\frac{\partial ^{2} f}{\partial x ^{2}} = - \frac{x}{( x + y ) ^{2}}

D (x, y) = \frac{- x ( x + 2 y ) - y ^{2}}{( x + y ) ^{4}}

Check the sign of at each critical point

Check critical point

(0, 1) :

Saddle

Saddle (0, 1)

f (x, y) = x^{2} + x y + y^{2} - 7 y + 16

f (x) = x^{2} (x - 2)^{2}

f (x) = \frac{2}{5} x^{5} - 5 x^{4} + 16 x^{3} + 2

f (x, y) = x^{2} + x y + y^{2} - 19 y + 120

f (x) = x^{4} + 8 x^{3}

What is the extreme f(x,y)=xln(x+y) ?
The extreme f(x,y)=xln(x+y) is Saddle(0,1)