Question

local extremes values ​​of ; ln (x ^ 2 + y ^ 2 +1)

Answer 1

Find derivative respect to x
Find derivative respect to y

Answer 2

\[f[x,y]\text{=}\text{Log}\left[1+x^2+y^2\right]\]
Derivative of f[x,y] with respect to x:
\[\frac{2 x}{1+x^2+y^2} \]
The same with respect to y:
\[\frac{2 y}{1+x^2+y^2} \]

Answer 3

Not exactly same, one is 2x, one is 2y. Set each to zero and solve for x and y (simultaneous equations). Solution x,y is critical point.

Answer 4

By inspection, both x and y are equal to zero and the point looks "extreme" in the everyday sense. Refer to the 3D plot.

Check the coordinates. The plot may be upside down because it looks better that way.