91Ó°ÊÓ

Assuming that the equations in Exercises \(25-28\) define \(y\) as a differentiable function of \(x,\) use Theorem 8 to find the value of \(d y / d x\) at the given point. $$ x y+y^{2}-3 x-3=0, \quad(-1,1) $$

Find all the local maxima, local minima, and saddle points of the functions in Exercises \(1-30\) . $$ f(x, y)=x^{4}+y^{4}+4 x y $$

In Exercises \(23-26,\) sketch the curve \(f(x, y)=c\) together with \(\nabla f\) and the tangent line at the given point. Then write an equation for the tangent line. $$ x^{2}-x y+y^{2}=7, \quad(-1,2) $$

Maximizing a product Find the largest product the positive numbers \(x, y,\) and \(z\) can have if \(x+y+z^{2}=16\) .

Display the values of the functions in Exercises \(19-28\) in two ways: (a) by sketching the surface \(z=f(x, y)\) and (b) by drawing an assortment of level curves in the function's domain. Label each level curve with its function value. $$ f(x, y)=4 x^{2}+y^{2}+1 $$

Find the limits in Exercises \(21-26\). $$ \lim _{P \rightarrow(0,-2,0)} \ln \sqrt{x^{2}+y^{2}+z^{2}} $$

Assuming that the equations in Exercises \(25-28\) define \(y\) as a differentiable function of \(x,\) use Theorem 8 to find the value of \(d y / d x\) at the given point. $$ x^{2}+x y+y^{2}-7=0, \quad(1,2) $$

Zero directional derivative In what direction is the derivative of \(f(x, y)=x y+y^{2}\) at \(P(3,2)\) equal to zero?

Rectangular box of longest volume in a sphere Find the dimensions of the closed rectangular box with maximum volume that can be inscribed in the unit sphere.

In Exercises \(25-30,\) find the linearization \(L(x, y)\) of the function at each point. $$ f(x, y)=3 x-4 y+5 \text { at } \quad \text { a. }(0,0), \quad \text { b. }(1,1) $$