Use a CAS to perform the following steps implementing the method of Lagrange multipliers for finding constrained extrema:

Evaluate f at each of the solution points and select the extreme value subject to the constraints asked for in the exercise.

Minimize $f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ subject to the constraints $x ^ { 2 } - x y + y ^ { 2 } - z ^ { 2 } - 1 = 0 \text { and } x ^ { 2 } + y ^ { 2 } - 1 = 0.$

$$\text { Find } \partial f / \partial x \text { and } \partial f / \partial y.$$

$$f ( x , y ) = ( x y - 1 ) ^ { 2 }$$

Find an equation for the plane tangent to the surface z = f(x, y) at the given point.

$$z = 1 / \left( x ^ { 2 } + y ^ { 2 } \right) , \quad ( 1,1,1 / 2 )$$

Use a CAS to perform the following steps implementing the method of Lagrange multipliers for finding constrained extrema:

Form the function $h = f - \lambda _ { 1 } g _ { 1 } - \lambda _ { 2 } g _ { 2 },$ where f is the function to optimize subject to the constraints $g_1 = 0\ and\ g_2 = 0.$

Minimize $f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ subject to the constraints $x ^ { 2 } - x y + y ^ { 2 } - z ^ { 2 } - 1 = 0 \text { and } x ^ { 2 } + y ^ { 2 } - 1 = 0.$

Find the point on the graph of

$$z = x ^ { 2 } + y ^ { 2 } + 10$$

nearest the plane x + 2y - z = 0.

Find the points on the curve

$$x y ^ { 2 } = 54$$

nearest the origin.

Sketch a typical level surface for the function. $f(x, y, z)=\ln \left(x^{2}+y^{2}+z^{2}\right)$

Find the domain and range of the given function and identify its level surfaces. Sketch a typical level surface. $f(x, y, z)=x^{2}+y^{2}-z$

Find the limit.

$$\lim _ { ( x , y ) \rightarrow ( 0 , \pi / 4 ) } \sec x \tan y$$

Show that the limits do not exist. $\lim _{(x, y) \rightarrow(1,0)} \frac{x e^{y}-1}{x e^{y}-1+y}$

Let T = f(x, y) be the temperature at the point (x, y) on the circle x = cos t, y = sin t, $0 \leq t \leq 2 \pi$ and suppose that $\frac { \partial T } { \partial x } = 8 x - 4 y , \quad \frac { \partial T } { \partial y } = 8 y - 4 x.$

Suppose that $T = 4 x ^ { 2 } - 4 x y + 4 y ^ { 2 }.$ Find the maximum and minimum values of T on the circle.

Find all the second-order partial derivatives of the functions. $g(x, y)=\cos x^{2}-\sin 3 y$

Find a. $\left(\frac{\partial w}{\partial y}\right)_{x}$ b. $\left(\frac{\partial w}{\partial y}\right)_{z}$ at the point (w, x, y, z) = (4, 2, 1, -1) if $w=x^{2} y^{2}+y z-z^{3}$ and $x^{2}+y^{2}+z^{2}=6.$

Use a CAS to perform the following steps implementing the method of Lagrange multipliers for finding constrained extrema:

Evaluate f at each of the solution points and select the extreme value subject to the constraints asked for in the exercise.

Maximize $f ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ subject to the constraints 2y + 4z - 5 = 0 and $4 x ^ { 2 } + 4 y ^ { 2 } - z ^ { 2 } = 0.$