What is the extreme 150(xy)^2+(0.5x+2y-2)^2 ?

The extreme 150(xy)^2+(0.5x+2y-2)^2 is Minimum(0,1),Minimum(4,0),Saddle(0.43685…,0.10921…)

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extreme 150(xy)^2+(0.5x+2y-2)^2

extreme points

150 (x y)^{2} + (0.5 x + 2 y - 2)^{2}

Minimum (0, 1), Minimum (4, 0), Saddle (0.43685 \dots, 0.10921 \dots)

Solution steps

Find the critical points:

(0, 1), (4, 0), (0.43685 \dots, 0.10921 \dots)

\frac{\partial ^{2} f}{\partial x ^{2}} = 4 (75 x^{2} + 2)

D (x, y) = 150 x^{2} - 270000 x^{2} y^{2} - 2400 x y + 2400 y^{2}

Check the sign of at each critical point

Check critical point

(0, 1) :

Minimum

Check critical point

(4, 0) :

Minimum

Check critical point

(0.43685 \dots, 0.10921 \dots) :

Saddle

Minimum (0, 1), Minimum (4, 0), Saddle (0.43685 \dots, 0.10921 \dots)

3 x^{2} - 5 x - 1

f (x, y) = x^{2} - 5 y x + 3 y^{2}

y = \frac{( 2 x + 3 )}{x + 2}

f (x) = \frac{x ^{2} - 5 x + 7}{x - 3}

f (x) = (3 x^{3} - 9)

What is the extreme 150(xy)^2+(0.5x+2y-2)^2 ?
The extreme 150(xy)^2+(0.5x+2y-2)^2 is Minimum(0,1),Minimum(4,0),Saddle(0.43685…,0.10921…)