What is the extreme f(a,b)=(9a^2+3b^2)e^{(ab)^2} ?

The extreme f(a,b)=(9a^2+3b^2)e^{(ab)^2} is Minimum(0,0)

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extreme f(a,b)=(9a^2+3b^2)e^{(ab)^2}

extreme points

f (a, b) = (9 a^{2} + 3 b^{2}) e^{(ab)^{2}}

Minimum (0, 0)

Solution steps

Find the critical points:

(0, 0)

\frac{\partial ^{2} f}{\partial a ^{2}} = 6 (6 e^{a^{2} b^{2}} a^{6} b^{2} + 2 e^{a^{2} b^{2}} a^{4} b^{4} + 5 e^{a^{2} b^{2}} a^{2} b^{2} + 3 e^{a^{2} b^{2}} a^{4} + e^{a^{2} b^{2}})

D (a, b) = 36 (2 e^{a^{2} b^{2}} a^{2} b^{6} + 6 e^{a^{2} b^{2}} a^{4} b^{4} + 15 e^{a^{2} b^{2}} a^{2} b^{2} + e^{a^{2} b^{2}} b^{4} + 3 e^{a^{2} b^{2}}) (6 e^{a^{2} b^{2}} a^{6} b^{2} + 2 e^{a^{2} b^{2}} a^{4} b^{4} + 5 e^{a^{2} b^{2}} a^{2} b^{2} + 3 e^{a^{2} b^{2}} a^{4} + e^{a^{2} b^{2}}) - 36 a^{2} (2 e^{a^{2} b^{2}} a^{2} b^{5} + 6 e^{a^{2} b^{2}} a^{4} b^{3} + 4 e^{a^{2} b^{2}} b^{3} + 12 e^{a^{2} b^{2}} a^{2} b)^{2}

Check the sign of at each critical point

Check critical point

(0, 0) :

Minimum

Minimum (0, 0)

- (x - 10) (x - 60)

f (x) = x^{4} - 24 x^{2} = x^{2} (x^{2} - 24)

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

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

f (x) = (\frac{1}{2} - x^{2} + y^{2}) e^{1 - x^{2} - y^{2}}

What is the extreme f(a,b)=(9a^2+3b^2)e^{(ab)^2} ?
The extreme f(a,b)=(9a^2+3b^2)e^{(ab)^2} is Minimum(0,0)