What is the extreme f(x)=8x-(8x^2)/2-xy^2+(8y^2)/(16)+7 ?

The extreme f(x)=8x-(8x^2)/2-xy^2+(8y^2)/(16)+7 is Maximum(1,0),Saddle(1/2 ,2),Saddle(1/2 ,-2)

extreme f(x)=8x-(8x^2)/2-xy^2+(8y^2)/(16)+7

extreme points

f (x) = 8 x - \frac{8 x ^{2}}{2} - x y^{2} + \frac{8 y ^{2}}{16} + 7

Maximum (1, 0), Saddle (\frac{1}{2}, 2), Saddle (\frac{1}{2}, - 2)

Solution steps

Find the critical points:

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

\frac{\partial ^{2} f}{\partial x ^{2}} = 1 - 2 x

D (x, y) = - 8 (1 - 2 x) - 4 y^{2}

Check the sign of at each critical point

Check critical point

(1, 0) :

Maximum

Check critical point

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

Saddle

Check critical point

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

Saddle

Maximum (1, 0), Saddle (\frac{1}{2}, 2), Saddle (\frac{1}{2}, - 2)

f (x, y) = x^{2} y + y^{3} - 27 y

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

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

g (x, y) = ln (x + y)

f (x, y) = - 4 x^{2} - 6 y^{2} - 4 x y + 800 x + 900 y - 10000

What is the extreme f(x)=8x-(8x^2)/2-xy^2+(8y^2)/(16)+7 ?
The extreme f(x)=8x-(8x^2)/2-xy^2+(8y^2)/(16)+7 is Maximum(1,0),Saddle(1/2 ,2),Saddle(1/2 ,-2)