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

The extreme f(x,y)=2ln(x)+4ln(y)-4x^2-y is Maximum(1/2 ,4),Maximum(-1/2 ,4)

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extreme f(x,y)=2ln(x)+4ln(y)-4x^2-y

extreme points

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

Maximum (\frac{1}{2}, 4), Maximum (- \frac{1}{2}, 4)

Solution steps

Find the critical points:

(\frac{1}{2}, 4), (- \frac{1}{2}, 4)

\frac{\partial ^{2} f}{\partial x ^{2}} = - \frac{4}{y ^{2}}

D (x, y) = - \frac{4 ( - 8 x ^{2} - 2 )}{x ^{2} y ^{2}}

Check the sign of at each critical point

Check critical point

(\frac{1}{2}, 4) :

Maximum

Check critical point

(- \frac{1}{2}, 4) :

Maximum

Maximum (\frac{1}{2}, 4), Maximum (- \frac{1}{2}, 4)

f (x) = \frac{1}{2} x^{4} - 4 x^{2} + 6

f (x) = x^{3} - 9 x^{2} + 15 x + 5

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

f (x) = 2 x^{2} + \frac{28}{x}

f (x) = ln (x^{2} + 3 x + 7), - 2 \leq x \leq 2

What is the extreme f(x,y)=2ln(x)+4ln(y)-4x^2-y ?
The extreme f(x,y)=2ln(x)+4ln(y)-4x^2-y is Maximum(1/2 ,4),Maximum(-1/2 ,4)