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)

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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) = \frac{2}{x ^{2} + 4}

f (x) = 2 x^{2} - x - 6

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

f (x) = x^{4} - 2 x^{2} + 5, - 2 \leq x \leq 2

\frac{x ^{\frac{1}{3}}}{x ^{2} + 8} - 10

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)