Chapter 10

For each problem, locate the critical points and classify each one using the second derivative test. a. \(f(x, y)=(x+y)^{2}\). b. \(f(x, y)=x^{2} y+x y^{2}\). c. \(f(x, y)=x^{4} y+x y^{4}-x y .\) d. \(f(x, y)=x^{2}-3 x y+2 x+10 y+6 y^{2}+12\). e. \(f(x, y)=\left(x^{2}-y^{2}\right) e^{-y}\)

For each problem, locate the critical points and evaluate the Hessian matrix at each critical point. a. \(f(x, y)=(x+y)^{2}\) b. \(f(x, y)=x^{2} y+x y^{2}\) c. \(f(x, y)=x^{4} y+x y^{4}-x y\) d. \(f(x, y, z)=x y+x z+y z\). e. \(f(x, y, z)=x^{2}+y^{2}+x z+2 z^{2}\)

Find the absolute maxima and minima of the function \(f(x, y)=x^{2}+\) \(x y+y^{2}\) on the unit circle.

A thin plate has a temperature distribution of \(T(x, y)=x^{2}-y^{3}-x^{2} y+\) \(y+20\) for \(0 \leq x, y, \leq 2\). Find the coldest and hottest points on the plate.

Find the extrema of the given function subject to the given constraint. a. \(f(x, y)=(x+y)^{2}, x^{2}+y=1\) b. \(f(x, y)=x^{2} y+x y^{2}, x^{2}+y^{2}=2\) c. \(f(x, y)=2 x+3 y, 3 x^{2}+2 y^{2}=3\) d. \(f(x, y, z)=x^{2}+y^{2}+z^{2}, x y z=1\) e. \(f(x, y, z)=x y+y x, x^{2}+y^{2}=1, x z=1\)

A particle moves under the force field \(F=-\nabla V\), where the potential function is given by \(V(x, y)=x^{3}+y^{3}-3 x y+5\). Find the equilibrium points of \(\mathbf{F}\) and determine if the equilibria are stable or unstable.

For each of the following, find a path that extremizes the given integral. a. \(f_{1}^{2}\left(y^{\prime 2}+2 y y^{\prime}+y^{2}\right) d y, y(1)=0, y(2)=1\). b. \(f_{0}^{2} y^{2}\left(1-y^{2}\right) d y, y(0)=1, y(2)=2\). c. \(f_{-1}^{1} 5 y^{\prime 2}+2 y y^{\prime} d y, y(-1)=1, y(1)=0\).

For each of the following, find a path that extremizes the given integral. a. \(\int_{1}^{2}\left(y^{\prime 2}+2 y y^{\prime}+y^{2}\right) d y, y(1)=0, y(2)=1\) b. \(\int_{0}^{3} y^{2}\left(1-y^{2}\right)^{2} d y, y(0)=1, y(2)=2\) c. \(\int_{-1}^{1} 5 y^{\prime 2}+2 y y^{\prime} d y, y(-1)=1, y(1)=0\)

A light ray travels from point A in a medium with index of refraction \(n_{1}\) toward point \(\mathrm{B}\) in a medium with index of refraction \(n_{2}\). Assume that the a. Write the time functional in terms of the travel path \(y(x)\). b. Apply Fermat's Principle of least time to write the Euler Equation for this functional. c. Solve the equation in part b. and show that \(n \sin \theta\) is a constant. d. Let the point at which the light is incident to the interface be at \((x, 0)\). Write an expression for the total time to travel from point \(A\) at \(\left(x_{1}, y_{1}\right)\) to point \(B\) at \(\left(x_{2}, y_{2}\right)\) in terms of the indices of refraction. and the coordinates \(x, x_{1}, x_{2}\), and \(y_{2}\). e. Treating the time as a function of \(x\), minimize this function as a function of one variable and derive Snell's law of refraction.

Given the cylinder defined by \(x^{2}+y^{2}=4\), find the path of shortest length connecting the given points. a. \((2,0,0)\) and \((0,2,5)\) b. \((2,0,0)\) and \((2,0,5)\)

The shape of a hanging chain between the points \((-a, b)\) and \((a, b)\) is such that the gravitational potential energy $$ V[y]=\rho g \int_{-a}^{a} y \sqrt{1+y^{\prime 2}} d x $$ is minimized subject to the length of the chain remaining constant, $$ L[y]=\int_{-a}^{a} \sqrt{1+y^{\prime 2}} d x $$ Find the shape, \(y(x)\) of the hanging chain.

Use a Lagrange multiplier to find the curve \(x(y)\) of length \(L=\pi\) on the interval \([0,1]\) which maximizes the integral \(I=\int_{0}^{1} y(x) d x\) and pass through the points \((0,0)\) and \((1,0)\).

A mass \(m\) lies on a table and is connected to a string of length \(\ell\) as shown in Figure 10.45. The string passes through a hole in the table and is connected to another mass \(M\) that is hanging in the air. We assume that the string remains taught and that mass \(M\) can only move vertically. a. The Lagrangian for this setup is $$ \mathcal{L}=\frac{1}{2} M \dot{r}^{2}+\frac{1}{2} m\left(\dot{r}^{2}+r^{2} \dot{\theta}^{2}\right)+M g(\ell-r) $$ where \(r\) and \(\theta\) are polar coordinates describing where mass \(m\) is on the table with respect to the hole. Explain why the terms in the Lagrangian are appropriate. b. Derive the equations of motion for \(r(t)\) and \(\theta(t)\). c. What angular velocity is needed for mass \(m\) to maintain uniform circular motion?