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

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

Mathematics
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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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

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Other answers:

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.

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