What is the extreme f(x,y)=5^2x^4+5^2y^4-4xy+14 ?

The extreme f(x,y)=5^2x^4+5^2y^4-4xy+14 is Saddle(0,0),Minimum(-1/5 ,-1/5),Minimum(1/5 , 1/5)

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extreme f(x,y)=5^2x^4+5^2y^4-4xy+14

extreme points

f (x, y) = 5^{2} x^{4} + 5^{2} y^{4} - 4 x y + 14

Saddle (0, 0), Minimum (- \frac{1}{5}, - \frac{1}{5}), Minimum (\frac{1}{5}, \frac{1}{5})

Solution steps

Find the critical points:

(0, 0), (- \frac{1}{5}, - \frac{1}{5}), (\frac{1}{5}, \frac{1}{5})

\frac{\partial ^{2} f}{\partial x ^{2}} = 300 y^{2}

D (x, y) = 90000 x^{2} y^{2} - 16

Check the sign of at each critical point

Check critical point

(0, 0) :

Saddle

Check critical point

(- \frac{1}{5}, - \frac{1}{5}) :

Minimum

Check critical point

(\frac{1}{5}, \frac{1}{5}) :

Minimum

Saddle (0, 0), Minimum (- \frac{1}{5}, - \frac{1}{5}), Minimum (\frac{1}{5}, \frac{1}{5})

f (x) = x^{3} - 3 x^{2} + x - 1

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

f (x) = e^{5 x}

f (x, y) = 2 x^{2} - 4 x + y^{2} - 6 y + 3

f (x) = x^{3} + 2, - 1 \leq x \leq 2

What is the extreme f(x,y)=5^2x^4+5^2y^4-4xy+14 ?
The extreme f(x,y)=5^2x^4+5^2y^4-4xy+14 is Saddle(0,0),Minimum(-1/5 ,-1/5),Minimum(1/5 , 1/5)