91Ó°ÊÓ

Each of these extreme value problems has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to ind the extreme values of the function subject to the given constraint. $$ f(x, y, z)=x^{4}+y^{4}+z^{4} ; \quad x^{2}+y^{2}+z^{2}=1 $$

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. $$ f(x, y)=x^{3}+y^{3}-3 x^{2}-3 y^{2}-9 x $$

Explain why the function is differentiable at the given point. Then find the linearization \(L(x, y)\) of the function at that point. \(f(x, y)=\sqrt{x y}, \quad(1,4)\)

Explain why the function is differentiable at the given point. Then find the linearization \(L(x, y)\) of the function at that point. \(f(x, y)=x^{2} e^{y}, \quad(1,0)\)

Let \(p(t)=f(g(t), h(t)),\) where \(f\) is differentiable, \(g(2)=4\) \(g^{\prime}(2)=-3, h(2)=5, h^{\prime}(2)=6, f_{x}(4,5)=2, f_{y}(4,5)=8\) Find \(p^{\prime}(2)\)

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. $$ f(x, y)=x^{4}-2 x^{2}+y^{3}-3 y $$

Find the limit, if it exists, or show that the limit does not exist. $$ \lim _{(x, y) \rightarrow(0,0)} \frac{x y}{\sqrt{x^{2}+y^{2}}} $$

Find and sketch the domain of the function. $$ f(x, y)=\sqrt{x-2}+\sqrt{y-1} $$

Each of these extreme value problems has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to ind the extreme values of the function subject to the given constraint. $$ f(x, y, z, t)=x+y+z+t ; \quad x^{2}+y^{2}+z^{2}+t^{2}=1 $$

Find the directional derivative of the function at the given point in the direction of the vector \(\mathbf{v} .\) $$ g(s, t)=s \sqrt{t}, \quad(2,4), \quad \mathbf{v}=2 \mathbf{i}-\mathbf{j} $$