What is the extreme y=x^2+z+z^2 ?

The extreme y=x^2+z+z^2 is Minimum(0,-1/2)

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extreme y=x^2+z+z^2

extreme points

y = x^{2} + z + z^{2}

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

Solution steps

Find the critical points:

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

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

D (x, z) = 4

Check the sign of at each critical point

Check critical point

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

Minimum

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

3 x^{3} - 3 x^{2} - 3 x + 8

2 x^{3} + 3 x^{2} - 12 x - 4

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

f (x) = x^{2} - 2 x - 1

f (x) = x y + \frac{8}{x} + \frac{8}{y}