What is the extreme f(x,y)=log_{10}(9x^2+16y^2-18x-64y-71) ?

The extreme f(x,y)=log_{10}(9x^2+16y^2-18x-64y-71) is Maximum(1,2)

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extreme f(x,y)=log_{10}(9x^2+16y^2-18x-64y-71)

extreme points

f (x, y) = lo g_{10} (9 x^{2} + 16 y^{2} - 18 x - 64 y - 71)

Maximum (1, 2)

Solution steps

Find the critical points:

(1, 2)

\frac{\partial ^{2} f}{\partial x ^{2}} = \frac{288 x ^{2} - 576 x + 2048 y - 512 y ^{2} - 6368}{ln ( 10 ) ( 9 x ^{2} + 16 y ^{2} - 18 x - 64 y - 71 ) ^{2}}

D (x, y) = \frac{- 46656 x ^{4} + 186624 x ^{3} + 663552 x ^{2} y - 165888 x ^{2} y ^{2} - 943488 x ^{2} + 331776 x y ^{2} + 1513728 x - 1327104 x y + 1179648 y ^{3} + 5382144 y + 8874432 - 147456 y ^{4} - 3704832 y ^{2}}{ln ^{2} ( 10 ) ( - 71 + 9 x ^{2} + 16 y ^{2} - 18 x - 64 y ) ^{4}}

Check the sign of at each critical point

Check critical point

(1, 2) :

Maximum

Maximum (1, 2)

x^{3} + y^{3} - 30 x y

f (x) = 1 - 9 x - 6 x^{2} - x^{3}

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

f (x, y) = x^{3} + 3 x y^{2} - 3 y^{2} + 3 x^{2} + 11

f (x) = - \frac{10}{x ^{2} - 3 x + 14}

What is the extreme f(x,y)=log_{10}(9x^2+16y^2-18x-64y-71) ?
The extreme f(x,y)=log_{10}(9x^2+16y^2-18x-64y-71) is Maximum(1,2)