What is the extreme f(x,y)=4^{x-y-7xy}+9y^2 ?

The extreme f(x,y)=4^{x-y-7xy}+9y^2 is Saddle((9\sqrt[7]{4}-7ln(2))/(49ln(2)), 1/7)

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extreme f(x,y)=4^{x-y-7xy}+9y^2

extreme points

f (x, y) = 4^{x - y - 7 x y} + 9 y^{2}

Solution steps

Find the critical points:

\frac{\partial ^{2} f}{\partial x ^{2}} = 2 (- ln^{2} (2) \cdot 2^{2 (x - y - 7 x y) + 1} (7 x + 1) (- 7 x - 1) + 9)

D (x, y) = 2401 ln^{4} (2) \cdot 2^{- 28 x y + 4 x - 4 y + 4} x^{2} y^{2} - 2401 ln^{4} (2) \cdot 4^{- 14 x y + 2 x - 2 y + 2} x^{2} y^{2} + 49 ln^{4} (2) \cdot 2^{- 28 x y + 4 x - 4 y + 4} y^{2} - 49 ln^{4} (2) \cdot 4^{- 14 x y + 2 x - 2 y + 2} y^{2} + 441 ln^{2} (2) \cdot 2^{- 14 x y + 2 x - 2 y + 3} y^{2} + 343 ln^{3} (2) \cdot 2^{- 28 x y + 4 x - 4 y + 4} x y + 49 ln^{3} (2) \cdot 2^{- 28 x y + 4 x - 4 y + 4} y - 63 ln^{2} (2) \cdot 2^{- 14 x y + 2 x - 2 y + 4} y + 49 ln^{4} (2) \cdot 2^{- 28 x y + 4 x - 4 y + 4} x^{2} - 49 ln^{4} (2) \cdot 4^{- 14 x y + 2 x - 2 y + 2} x^{2} - 49 ln^{3} (2) \cdot 2^{- 28 x y + 4 x - 4 y + 4} x + ln^{4} (2) \cdot 2^{- 28 x y + 4 x - 4 y + 4} - ln^{4} (2) \cdot 4^{- 14 x y + 2 x - 2 y + 2} - 7 ln^{3} (2) \cdot 2^{- 28 x y + 4 x - 4 y + 4} + 9 ln^{2} (2) \cdot 2^{- 14 x y + 2 x - 2 y + 3} - 49 ln^{2} (2) \cdot 4^{2 (- 7 x y + x - y) + 1}

Check the sign of at each critical point

Check critical point

Saddle

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

f (x, y) = (\frac{2}{6 ^{- x^{2} - y^{2} + 2}})

f (x, y) = 9 x^{2} + 9 x y + 6 y^{2} + 4 x y^{2} + 6 x^{2} y

f (x) = x^{\frac{6}{7}}, - 2 \leq x \leq 4

f (x, y) = \frac{1}{e ^{x - y}}

What is the extreme f(x,y)=4^{x-y-7xy}+9y^2 ?
The extreme f(x,y)=4^{x-y-7xy}+9y^2 is Saddle((9\sqrt[7]{4}-7ln(2))/(49ln(2)), 1/7)