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\(f(x,y)=\ln (\sec (x^{2}+y^{2}))\) at \(\left( \sqrt{\dfrac{\pi }{8}},\sqrt{\dfrac{\pi }{8}}\right) ; \mathbf{a}=\mathbf{i}+\mathbf{j}\)

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\(f(x,y)=\dfrac{x^{2}-y^{2}}{x^{2}+y^{2}}\) at \((1,1); \mathbf{a}=\mathbf{i}-\mathbf{j}\)

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\(f(x,y)=y\sin ^{-1}x\) at \((0,1)\); \(\mathbf{a}=3\mathbf{i}+\mathbf{j}\)

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\(f(x,y,z)=\ln (xyz)\) at \((1,1,3);\) \(\mathbf{a}=\mathbf{i}+\mathbf{j}+3\mathbf{k}\)

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\(f(x,y,z)=ye^{-x}(x^{2}+y^{2}+z^{2}+1)\) at \((0,0,0);\) \(\mathbf{a}=2\mathbf{i}+\mathbf{j}+2\mathbf{k}\)

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\(f(x,y,z)=(2x+y+z)^{2}+xyz\) at \((1,1,1);\) \(\mathbf{a}=-\mathbf{i}-\mathbf{j}-\mathbf{k}\)

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\(f(x,y)=\sec (x^{2}+y^{2})\) at \(\left( \sqrt{\dfrac{\pi }{8}},\sqrt{\dfrac{\pi }{8}}\right) \)

Question 8

\(f(x,y)=\dfrac{x^{2}-y^{2}}{x^{2}+y^{2}}\) at \((1,1) \)

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\(f(x,y)=y\sin ^{-1}x\) at \((0,1)\)

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\(x^{2}-y^{2}+z^{2}=4\) at \((-1,1,2)\)

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\(2x^{2}+y^{2}=z\) at \((1,0,2)\)

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\(f(x,y)=y\sin ^{-1}x\) at \(( 0,1) \)

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\(z=f(x,y)=x^{2}+xy+y^{2}+6x\)

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\(z= f(x,y)=x^{3}-y^{3}+3xy\)

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\(z=f( x,y) =xy\)

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\(z=f( x,y) =xe^{xy}\)

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\(z=f( x,y) =\sin x+\sin y\)

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\(z=f( x,y) =x^{2}-9y+y^{2}\)

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\(z=f( x,y) = x^{2}- 2xy +2y\), \(D\): \(0\le x \le 3, 0\le y\le 4\).

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\(z=f( x,y) =3xy^{2}\), \(D\): \(x^{2}+y^{2}\leq 9\).

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\(z=f( x,y) =5x^{2}+3y^{2}+xy\), \(g( x,y) =2x-y-20=0\)

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\(z=f( x,y) =x\sqrt{y}\), \(g( x,y) =2x+y-3000=0\)

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\(z=f( x,y) =x^{2}+y^{2}\), \(g( x,y) =2x+y-4=0\)

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\(z=f( x,y) =xy^{2}\), \(g( x,y)=x^{2}+y^{2}-1=0\)

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Heat Transfer A metal plate is placed on the \(xy\)-plane in such a way that the temperature \(T\) at any point \((x,y)\) is given by \(T = e^{y}(\sin x + \sin y)^{\circ}\) C.

1. What is the rate of change in temperature at \((0,0)\) in the direction of \(4\mathbf{i} + 3\mathbf{j}\)?
2. At \((0,0)\), in what direction is the temperature increasing most rapidly?
3. In what direction is the temperature decreasing most rapidly?
4. In what direction does the temperature remain the same?

Question

Maximizing Profit A company produces two products at a total cost \(C( x,y) =x^{2}+200x+y^{2}+100y-xy,\) where \(x\) and \(y\) represent the units of each product sold. The revenue function is \(R( x,y) =2000x-2x^{2}+100y-y^{2}+xy\).

1. Find the number of units of each product that will maximize profit.
2. What is the maximum profit?

901

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Maximizing Profit A manufacturer introduces a new product with a Cobb–Douglas production function of \(P( x,y) = 10K^{0.3}L^{0.7},\) where \(K\) represents the units of capital and \(L\) the units of labor needed to produce \(P\) units of the product. A total of $51,000 has been budgeted for production. Each unit of labor costs the manufacturer $100 and each unit of capital costs $50.

1. How should the $51,000 be allocated between labor and capital to maximize production?
2. What is the maximum number of units that can be produced?

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Volume Find the volume of the largest rectangular solid that can be inscribed in the interior of the surface \[ \dfrac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1 \] if the sides of the solid are parallel to the axes.

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Use Lagrange multipliers to find the points on the surface \(xyz=1\) closest to the origin.

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Volume Use Lagrange multipliers to maximize the volume of a rectangular solid that has three faces on the coordinate planes and one vertex on the plane \(\dfrac{x}{a} +\dfrac{y}{b} +\dfrac{z}{c} =1,\)\(a>0,\)\(b>0,\)\(c>0.\)

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Minimizing Cost The base of a rectangular box costs five times as much as do the other five sides. Use Lagrange multipliers to find the proportions of the dimensions for the cheapest possible box of volume \(V\).

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Find the extreme values of \(f(x,y,z)=xyz\) subject to the constraints \(x^{2}+y^{2}=1\) and \(y=3z\).