What is the extreme g(x,y)=-x^2+xy-25/512 y^4-7 ?

The extreme g(x,y)=-x^2+xy-25/512 y^4-7 is Saddle(0,0),Maximum(-4/5 ,-8/5),Maximum(4/5 , 8/5)

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extreme g(x,y)=-x^2+xy-25/512 y^4-7

extreme points

g (x, y) = - x^{2} + x y - \frac{25}{512} y^{4} - 7

Saddle (0, 0), Maximum (- \frac{4}{5}, - \frac{8}{5}), Maximum (\frac{4}{5}, \frac{8}{5})

Solution steps

Find the critical points:

(0, 0), (- \frac{4}{5}, - \frac{8}{5}), (\frac{4}{5}, \frac{8}{5})

\frac{\partial ^{2} f}{\partial x ^{2}} = - \frac{75 y ^{2}}{128}

D (x, y) = \frac{75 y ^{2}}{64} - 1

Check the sign of at each critical point

Check critical point

(0, 0) :

Saddle

Check critical point

(- \frac{4}{5}, - \frac{8}{5}) :

Maximum

Check critical point

(\frac{4}{5}, \frac{8}{5}) :

Maximum

Saddle (0, 0), Maximum (- \frac{4}{5}, - \frac{8}{5}), Maximum (\frac{4}{5}, \frac{8}{5})

f (x) = 2 x^{3} - 30 x^{2} + 144 x - 10

f (t) = 64 - x^{2}, - 8 \leq x \leq 8

f (x, y) = 4 x^{2} - 8 x y + 2 y^{2}

f (x) = - 3 x^{2} + 24 x - 200

2 - x^{4} + 2 x^{2} - y^{2}

What is the extreme g(x,y)=-x^2+xy-25/512 y^4-7 ?
The extreme g(x,y)=-x^2+xy-25/512 y^4-7 is Saddle(0,0),Maximum(-4/5 ,-8/5),Maximum(4/5 , 8/5)