**Group 1 Question**

The set of all points forms a sphere-like object called an ellipsoid centered at the point (a picture of this surface is shown below).

Find the points on this ellipsoid that are closest to and farthest away from the point the origin.

**Group 2 Question**

Find the equation for the tangent plane to the surface determined by

at the point .

**Group 3 Question**

Find the maximum and minimum values of over all points in the region of space .

**Group 4 Question**

Find the linear approximation of the function

at the point $latex (0, 1)4.

**Group 5 Question**

Let be a real-valued function that satisfies

(and note that there are many functions that satisfy this). Given this information, compute the partial derivative at the point with spherical coordinates .

**Group 6 Question**

For this question let be a symmetric, matrix of real numbers

and define a function by

.

(a) Show that the gradient of satisfies .

(b) Compute the Hessian matrix of .

(c) Show that has a critical point \textit{on the unit circle} precisely when

for some .

(d) (Bonus) What is a vector that satisfies an equation like that in part (c) called?

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