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Chapter 15: Problem 11

The function has a critical point at (0,0) What sort of critical point is it? $$h(x, y)=\cos x \cos y$$

Short Answer

Expert verified
The critical point at (0,0) is a local maximum.

Step by step solution

01

Understand the function

The given function is \( h(x, y) = \cos x \cos y \). We need to determine the nature of the critical point at \((0,0)\).
02

Compute partial derivatives

First, we calculate the first partial derivatives of the function:- \( h_x = \frac{\partial}{\partial x} (\cos x \cos y) = -\sin x \cos y \)- \( h_y = \frac{\partial}{\partial y} (\cos x \cos y) = \cos x (-\sin y) = -\cos x \sin y \) Both partial derivatives are zero at the point \((0,0)\) because \( \sin 0 = 0 \) and \( \cos 0 = 1 \).
03

Compute second partial derivatives

We now find the second partial derivatives needed for the Hessian determinant:- \( h_{xx} = \frac{\partial}{\partial x} (-\sin x \cos y) = -\cos x \cos y \)- \( h_{yy} = \frac{\partial}{\partial y} (-\cos x \sin y) = -\cos x \cos y \)- \( h_{xy} = \frac{\partial}{\partial y} (-\sin x \cos y) = \sin x \sin y \)At \((0,0)\), \( h_{xx} = -1 \), \( h_{yy} = -1 \), and \( h_{xy} = 0 \).
04

Calculate the Hessian determinant

The Hessian determinant \( D \) is calculated as:\[ D = h_{xx} h_{yy} - (h_{xy})^2 \]Substitute the values:\[ D = (-1)(-1) - (0)^2 = 1 \]
05

Classify the critical point

Since \( D > 0 \) and \( h_{xx} < 0 \) at the point \((0,0)\), the function has a local maximum at this critical point. This follows from the criteria for classifying critical points using the Hessian determinant.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Partial Derivatives
Partial derivatives help us understand how a multivariable function changes with respect to one variable at a time. For the given function \( h(x, y) = \cos x \cos y \), we first calculate \( h_x \), the partial derivative with respect to \( x \):
  • We treat \( y \) as a constant and differentiate \( \cos x \), resulting in \( -\sin x \).
  • So, \( h_x = -\sin x \cos y \).
Similarly, the partial derivative with respect to \( y \), \( h_y \), is calculated by treating \( x \) as a constant:
  • We differentiate \( \cos y \) as \( -\sin y \).
  • Thus, \( h_y = -\cos x \sin y \).
Both \( h_x \) and \( h_y \) must equal zero at a critical point. At \((0,0)\), these reduce to zero since \( \sin 0 = 0 \) and \( \cos 0 = 1 \). Thus, partial derivatives provide crucial information for pinpointing critical points.
Hessian Determinant
The Hessian determinant gives insights into the nature of a critical point, assessing whether it is a local maximum, minimum, or saddle point. The Hessian matrix consists of second partial derivatives:
  • \( h_{xx} = \frac{\partial^2}{\partial x^2} (\cos x \cos y) = -\cos x \cos y \)
  • \( h_{yy} = \frac{\partial^2}{\partial y^2} (\cos x \cos y) = -\cos x \cos y \)
  • \( h_{xy} = \frac{\partial^2}{\partial y \partial x} (\cos x \cos y) = \sin x \sin y \)
At point \((0,0)\), these values are \( h_{xx} = -1 \), \( h_{yy} = -1 \), and \( h_{xy} = 0 \).
The Hessian determinant \( D \) is calculated as:\[ D = h_{xx} h_{yy} - (h_{xy})^2 \]Substituting values:\[ D = (-1)(-1) - (0)^2 = 1 \]A positive \( D \) indicates that we may have a local maximum or minimum.
Local Maximum
To determine the type of critical point, we look at the Hessian determinant and the sign of \( h_{xx} \). If \( D > 0 \) and \( h_{xx} < 0 \), the function has a local maximum at the critical point.
In this exercise, at the critical point \((0,0)\):
  • \( D = 1 \), which is greater than zero.
  • \( h_{xx} = -1 \), which is less than zero.
Therefore, these values satisfy the conditions for a local maximum.
A local maximum means that around the critical point \((0,0)\), the function \( h(x, y) \) does not reach a higher value, producing a peak-like shape on the graph.

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Most popular questions from this chapter

Use Lagrange multipliers to find the maximum and minimum values of \(f\) subject to the given constraint, if such values exist. $$f(x, y)=x y, \quad x^{2}+2 y^{2} \leq 1$$

(a) If \(\sum_{i=1}^{3} x_{i}=1,\) find the values of \(x_{1}, x_{2}, x_{3}\) making \(\sum_{i=1}^{3} x_{i}^{2}\) minimum. (b) Generalize the result of part (a) to find the minimum value of \(\sum_{i=1}^{n} x_{i}^{2}\) subject to \(\sum_{i=1}^{n} x_{i}=1\)

The Dorfman-Steiner rule shows how a company which has a monopoly should set the price, \(p,\) of its product and how much advertising, \(a\), it should buy. The price of advertising is \(p_{a}\) per unit. The quantity, \(q\), of the product sold is given by \(q=K p^{-E} a^{\theta},\) where \(K>0, E>1\) and \(0<\theta<1\) are constants. The cost to the company to make each item is \(c\) (a) How does the quantity sold, \(q\), change if the price, \(p,\) increases? If the quantity of advertising, \(a,\) increases? (b) Show that the partial derivatives can be written in the form \(\partial q / \partial p=-E q / p\) and \(\partial q / \partial a=\theta q / a\). (c) Explain why profit, \(\pi\), is given by \(\pi=p q-c q-p_{a} a\). (d) If the company wants to maximize profit, what must be true of the partial derivatives, \(\partial \pi / \partial p\) and \(\partial \pi / \partial a ?\) (e) Find \(\partial \pi / \partial p\) and \(\partial \pi / \partial a\). (i) Use your answers to parts (d) and (e) to show that at maximum profit, $$\frac{p-c}{p}=\frac{1}{E} \quad \text { and } \quad \frac{p-c}{p_{a}}=\frac{a}{\theta q}$$ (g) By dividing your answers in part ( \(f\) ), show that at maximum profit, $$\frac{p_{a} a}{p q}=\frac{\theta}{E}$$ This is the Dorfman-Steiner rule, that the ratio of the advertising budget to revenue does not depend on the price of advertising.

Let \(f(x, y)=x^{3}+k y^{2}-5 x y .\) Determine the values of \(k\) (if any) for which the critical point at (0,0) is: (a) A saddle point (b) A local maximum (c) A local minimum

Each person tries to balance his or her time between leisure and work. The tradeoff is that as you work less your income falls. Therefore each person has indiffer. ence curves which connect the number of hours of leisure, \(l\), and income, \(s\). If, for example, you are indifferent between 0 hours of leisure and an income of \(\$ 1125\) a week on the one hand, and 10 hours of leisure and an income of \(\$ 750\) a week on the other hand, then the points \(l=0, s=1125,\) and \(l=10, s=750\) both lie on the same indifference curve. Table 15.3 gives information on three indifference curves, 1 , II, and III. $$\begin{array}{c|c|c|c|c|c} \hline \text { I } & \text { II } & \text { III } & \text { I } & \text { II } & \text { III } \\ \hline 1125 & 1250 & 1375 & 0 & 20 & 40 \\ \hline 750 & 875 & 1000 & 10 & 30 & 50 \\ \hline 500 & 625 & 750 & 20 & 40 & 60 \\ \hline 375 & 500 & 625 & 30 & 50 & 70 \\ \hline 250 & 375 & 500 & 50 & 70 & 90 \\ \hline \end{array}$$ (a) Graph the three indifference curves. (b) You have 100 hours a weck available for work and leisure combined, and you earn S 10/hour. Write an equation in terms of \(l\) and \(s\) which represents this constraint. (c) On the same axes, graph this constraint. (d) Estimate from the graph what combination of leisure hours and income you would choose under these circumstances. Give the corresponding number of hours per week you would work.
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