By Henry Helson

ISBN-10: 0201127520

ISBN-13: 9780201127522

This e-book can be of curiosity to post-graduates on classes in harmonic research.

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**Sample text**

MULTIVARIABLE OPTIMIZATION Substituting back into Eq. 2), we obtain y = 532, 308 at the maximum. 11 shows a Maple graph of the level sets of f together with the feasible region. The level sets f = C for C = 0, 100, 000, . . , 500, 000 form smaller and smaller concentric rings, all of which intersect the feasible region. The level set f = 532, 308 forms the smallest ring. This set barely touches the feasible region S, and is tangent to the line x1 + x2 = 10, 000 at the optimum point. This graphical evidence indicates that the critical point found by using Lagrange multipliers along the constraint line x1 + x2 = 10, 000 is actually the maximum of the function f over the feasible region S.

To compute S(y, a) we will need to obtain a formula for dy/da. One way to do this would be to employ one–variable methods directly on Eq. 10), perhaps with the aid of a computer algebra system. Another method, which is computationally more eﬃcient, is to use the multivariable chain rule, which implies that dy ∂y dx1 ∂y dx2 ∂y = + + . 11) da ∂x1 da ∂x2 da ∂a Since both ∂y/∂x1 and ∂y/∂x2 are zero at the optimum, we have ∂y dy = = −x21 da ∂a directly from Eq. 40. 947, 349 Thus, a 10% increase in price elasticity for 19–inch sets will result in a 4% drop in proﬁt.

S. What is the minimum tariﬀ that would make it worthwhile for the company to relocate its facility? (d) Given that the tariﬀ is large enough to motivate the company to move its facility, how important is the actual tariﬀ amount? Consider the sensitivity of both production levels and proﬁt to the amount of the tariﬀ. 6. A manufacturer of personal computers currently sells 10,000 units per month of a basic model. The cost of manufacture is $700/unit, and the wholesale price is $950. During the last quarter the manufacturer lowered the price $100 in a few test markets, and the result was a 50% increase in sales.

### Harmonic Analysis by Henry Helson

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