What is the extreme yln(x)+xy^2 ?

The extreme yln(x)+xy^2 is Minimum(-1/(e^2),e^2),Saddle(0,1)

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extreme yln(x)+xy^2

extreme points

y ln (x) + x y^{2}

Minimum (- \frac{1}{e ^{2}}, e^{2}), Saddle (0, 1)

Solution steps

Find the critical points:

(- \frac{1}{e ^{2}}, e^{2}), (0, 1)

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

D (y, x) = \frac{- 4 y ^{2} x ^{2} - 6 y x - 1}{x ^{2}}

Check the sign of at each critical point

Check critical point

(- \frac{1}{e ^{2}}, e^{2}) :

Minimum

Check critical point

(0, 1) :

Saddle

Minimum (- \frac{1}{e ^{2}}, e^{2}), Saddle (0, 1)

2 x^{3} - 9 x^{2} + 12 x - 8

f (x) = 4 x^{2} - x y + 2 y^{2}

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

f (x) = 2 x + 5 x^{- 1}

a - \frac{ab}{( a + b )}

What is the extreme yln(x)+xy^2 ?
The extreme yln(x)+xy^2 is Minimum(-1/(e^2),e^2),Saddle(0,1)