91Ó°ÊÓ

(a) Use Lagrange multipliers to prove that the product of three positive numbers \(x, y\), and \(z\), whose sum has the constant value \(S\), is a maximum when the three numbers are equal. Use this result to prove that \(\sqrt[3]{x y z} \leq \frac{x+y+z}{3}\). (b) Generalize the result of part (a) to prove that the product \(x_{1} x_{2} x_{3} \cdot \cdots x_{n}\) is a maximum when \(x_{1}=x_{2}=x_{3}=\) \(\cdots=x_{n}, \sum_{i=1}^{n} x_{i}=S\), and all \(x_{i} \geq 0 .\) Then prove that $$ \sqrt[n]{x_{1} x_{2} x_{3} \cdot \cdots x_{n}} \leq \frac{x_{1}+x_{2}+x_{3}+\cdots \cdot+x_{n}}{n}. $$ This shows that the geometric mean is never greater than the arithmetic mean.

State the Second Partials Test for relative extrema and saddle points.

Let \(T(x, y, z)=100+x^{2}+y^{2}\) represent the temperature at each point on the sphere \(x^{2}+y^{2}+z^{2}=50 .\) Find the maximum temperature on the curve formed by the intersection of the sphere and the plane \(x-z=0\)

Use the result of Exercise 39 to find the least squares regression quadratic for the given points. Use the regression capabilities of a graphing utility to confirm your results. Use the graphing utility to plot the points and graph the least squares regression quadratic. $$ (-2,0),(-1,0),(0,1),(1,2),(2,5) $$

Find symmetric equations of the tangent line to the curve of intersection of the surfaces at the given point, and (b) find the cosine of the angle between the gradient vectors at this point. State whether or not the surfaces are orthogonal at the point of intersection.\(z=x^{2}+y^{2}, \quad z=4-y, \quad(2,-1,5)\)

The function \(f\) is homogeneous of degree \(n\) if \(f(t x, t y)=t^{n} f(x, y) .\) Determine the degree of the homogeneous function, and show that \(x f_{x}(x, y)+y f_{y}(x, y)=n f(x, y)\). $$ f(x, y)=x^{3}-3 x y^{2}+y^{3} $$

Find symmetric equations of the tangent line to the curve of intersection of the surfaces at the given point, and (b) find the cosine of the angle between the gradient vectors at this point. State whether or not the surfaces are orthogonal at the point of intersection.\(z=\sqrt{x^{2}+y^{2}}, \quad 5 x-2 y+3 z=22, \quad(3,4,5)\)

A function \(f\) has continuous second partial derivatives on an open region containing the critical point \((a, b)\). If \(f_{x x}(a, b)\) and \(f_{y y}(a, b)\) have opposite signs, what is implied? Explain.

Consider the objective function \(g(\alpha, \beta, \gamma)=\) \(\cos \alpha \cos \beta \cos \gamma\) subject to the constraint that \(\alpha, \beta\), and \(\gamma\) are the angles of a triangle. (a) Use Lagrange multipliers to maximize \(g\). (b) Use the constraint to reduce the function \(g\) to a function of two independent variables. Use a computer algebra system to graph the surface represented by \(g .\) Identify the maximum values on the graph.

The table shows the world populations \(y\) (in billions) for five different years. (Source: U.S. Bureau of the Census, International Data Base) $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { Year } & 1994 & 1996 & 1998 & 2000 & 2002 \\ \hline \text { Population, } \boldsymbol{y} & 5.6 & 5.8 & 5.9 & 6.1 & 6.2 \\ \hline \end{array} $$ Let \(x=4\) represent the year 1994 . (a) Use the regression capabilities of a graphing utility to find the least squares regression line for the data. (b) Use the regression capabilities of a graphing utility to find the least squares regression quadratic for the data. (c) Use a graphing utility to plot the data and graph the models. (d) Use both models to forecast the world population for the year \(2010 .\) How do the two models differ as you extrapolate into the future?