1. Mathematics
2. Calculus and Analysis
3. Real Analysis
4. Integrals
5. 22467

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The problems are attached

1 -5 based on Chapter Partial Derivative - (Maximum & Minimum Values and Lagrange Multipliers

1. Locate all relative maxima, relative minima, and saddle points of the surface defined by the following function.

2. Consider the minimization of
subject to the constraint of

(a) Draw the constraint curve on top of with y-axis
and x-axis between -2 and 6. Estimate where extrema values may occur and
compute the function values corresponding to these extrema.

(b) Solve the problem in part (a) with the aid Lagrange multipliers. You may
have to solve the equations numerically. Compare your answers with those in
the part (a).

3. Let represent the temperature at
each of the sphere . Find the maximum
temperature on the curve formed by the intersection of the sphere and the
.

Apply the method of Lagrange multipliers with two constraints. That is, maximize
where and where

4. Find the absolute extrema of the following function on the indicated closed and bounded set R.

5. An international organization must decide how to spend the $2000 they have been
allotted for famine relief in a remote area. They expect to divide the money between buying rice at $5/sack and beans at $10/sack. The number, P, people who would be fed if they buy x sacks of rice and y sacks of beans is given by

See attachments for proper equations and questions