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Problem 1

For each function, evaluate (a) \(f(0,0)\); (b) \(f(1,0) ;\) (c) \(f(0,-1)\); (d) \(f(a, 2) ;\) (e) \(f(y, x)\);(f) \(f(x+h, y+k)\) HINT [See Quick Examples page 1080.] $$ f(x, y)=x^{2}+y^{2}-x+1 $$

Expert verified

The short answers for the given function, \(f(x,y) = x^2 + y^2 - x + 1\), evaluated at different inputs are:
(a) \(f(0,0) = 1\)
(b) \(f(1,0) = 1\)
(c) \(f(0,-1) = 2\)
(d) \(f(a,2) = a^2 - a + 5\)
(e) \(f(y,x) = y^2 + x^2 - y + 1\)
(f) \(f(x+h, y+k) = x^2 + 2xh + h^2 + y^2 + 2yk + k^2 - x - h + 1\)

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Chapter 15

Sketch the graph of a function that has one extremum and no saddle points.

Chapter 15

The Gym Shirt Company manufactures cotton socks. Production is partially automated through the use of robots. Daily operating costs amount to $$\$ 150$$ per laborer and $$\$ 60$$ per robot. The number of pairs of socks the company can manufacture in a day is given by a Cobb-Douglas production formula $$ q=50 n^{0.6} r^{0.4} $$ where \(q\) is the number of pairs of socks that can be manufactured by \(n\) laborers and \(r\) robots. Assuming that the company has a daily operating budget of $$\$ 1,500$$ and wishes to maximize productivity, how many laborers and how many robots should it use? What is the productivity at these levels? HINT [See Example 5.]

Chapter 15

Find the point on the plane \(2 x-2 y-z+1=0\) closest to \((1,1,0)\).

Chapter 15

Let \(H=f_{x x}(a, b) f_{y y}(a, b)-f_{x y}(a, b)^{2} .\) What condition on \(H\) guarantees that \(f\) has a relative extremum at the point \((a, b) ?\)

Chapter 15

Locate and classify all the critical points of the functions. HINT [See Example 2.] $$ f(x, y)=x y+\frac{2}{x}+\frac{2}{y} $$

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