91Ó°ÊÓ

Solve the given optimization problem by using substitution. HINT [See Example 1.] Find the maximum value of \(f(x, y, z)=1-x^{2}-x-y^{2}+\) \(y-z^{2}+z\) subject to \(3 x=y\). Also find the corresponding \(\operatorname{point}(\mathrm{s})(x, y, z)\)

Classify the shaded value in each table as one of the following: a. a relative maximum b. a relative minimum c. a saddle point d. neither a relative extremum nor a saddle point $$ \begin{array}{|r|r|r|r|r|r|r|} \hline & -3 & -2 & -1 & 0 & 1 & 2 \\ \hline \mathbf{- 3} & 2 & 3 & 2 & -1 & -6 & -13 \\ \hline \mathbf{- 2} & 3 & 4 & 3 & 0 & -5 & -12 \\ \hline \mathbf{- 1} & 2 & 3 & 2 & -1 & -6 & -13 \\ \hline \mathbf{0} & -1 & 0 & -1 & -4 & -9 & -16 \\ \hline \mathbf{1} & -6 & -5 & -6 & -9 & -14 & -21 \\ \hline \mathbf{2} & -13 & -12 & -13 & -16 & -21 & -28 \\ \hline \mathbf{3} & -22 & -21 & -22 & -25 & -30 & -37 \\ \hline \end{array} $$

At what points on the sphere \(x^{2}+y^{2}+z^{2}=1\) is the product \(x y z\) a maximum? (The method of Lagrange multipliers can be used.)

Your latest CD-ROM of clip art is expected to sell between \(q=8,000-p^{2}\) and \(q=10,000-p^{2}\) copies if priced at \(p\) dollars. You plan to set the price between $$\$ 40$$ and $$\$ 50 .$$ What is the average of all the possible revenues you can make? HINT [See Example 4.

Package Dimensions: USPS The U.S. Postal Service (USPS) will accept only packages with a length plus girth no more than 108 inches. \({ }^{26}\) (See the figure.) What are the dimensions of the largest volume package that the USPS will accept? What is its volume?

Sketch the level curves \(f(x, y)=c\) for the given function and values of c. HINT [See Example 5.] $$ f(x, y)=y+2 x^{2} ; c=-2,0,2 $$

Your weekly cost (in dollars) to manufacture \(x\) cars and \(y\) trucks is \(C(x, y)=200,000+6,000 x+4,000 y-100,000 e^{-0.01(x+y)} .\) What is the marginal cost of a car? Of a truck? How do these marginal costs behave as total production increases?

Recall that the compound interest formula for annual compounding is $$ A(P, r, t)=P(1+r)^{t} $$ where \(A\) is the future value of an investment of \(P\) dollars after \(t\) years at an interest rate of \(r\). a. Calculate \(\frac{\partial A}{\partial P}, \frac{\partial A}{\partial r}\), and \(\frac{\partial A}{\partial t}\), all evaluated at \((100,0.10,10)\). (Round your answers to two decimal places.) Interpret your answers. b. What does the function \(\left.\frac{\partial A}{\partial P}\right|_{(100,0.10, t)}\) of \(t\) tell about your investment?

Your weekly cost (in dollars) to manufacture \(x\) cars and \(y\) trucks is $$ C(x, y)=240,000+6,000 x+4,000 y $$ a. What is the marginal cost of a car? Of a truck? HINT [See Example 1.] b. Describe the graph of the cost function C. HINT [See Example 7.] c. Describe the slice \(x=10\). What cost function does this slice describe? d. Describe the level curve \(z=480,000\). What does this curve tell you about costs?

Your sales of online video and audio clips are booming. Your Internet provider, Moneydrain.com, wants to get in on the action and has offered you unlimited technical assistance and consulting if you agree to pay Moneydrain \(3 \&\) for every video clip and \(4 \&\) for every audio clip you sell on the site. Further, Moneydrain agrees to charge you only $$\$ 10$$ per month to host your site. Set up a (monthly) cost function for the scenario, and describe each variable.