Example of inflection point-differential equation, Mathematics

Example of inflection point

Determine the points of inflection on the curve of the function

y = x3

Solution

The only possible inflexion points will happen where

(d2y)/( dx2)   = 0

From specified function as:

(dy)/(dx) = 3x2 and (d2y)/( dx2)   = 6x

Equating the second derivative to zero, we include

 6x = 0 or x = 0

We test whether the point at that x = 0 is an inflexion point as follows

While x is slightly less than 0, ((d2y)/(dx2)) < 0; it means a downward concavity

While x is slightly larger than 0, ((d2y)/(dx2)) > 0;  it means an upward concavity

Hence we have a point of inflexion at point x = 0 since the concavity of the curve changes as we pass from the left to the right of x = 0

Illustration:-

                                 924_Example of inflection point.png

 

maximize and the revenue

1. The per week revenue Sh. R of a small company is specified by

 R = (14 + 81x - (x3/12)) whereas x is the number of units produced.

Required

i.          Find out the number of units that maximize the revenue

ii.         Find out the maximum revenue

iii.        Find out the price per unit that will maximize revenue

Solution

i. To determine maximum or minimum value we needs differential calculus as given below:

R = (14 + 81x - (x3/12))

(dR)/(dx) = 81 - (1/12) . (3x2)

(d2R)/(dx2) = 0 - (1/12) . (3.2x) = -(x/2)

Put (dR)/(dx) = 0 that is 81 - (1/4)x2 = 0

That gives x = 18 or x = -18

(d2R)/(dx2) = -(x/2)

Hence when x = 18;

(d2R)/(dx2) = -9

That is (-) negative indicating a maximum value.

Hence at x = 18, the value of R is a maximum. Correspondingly at x = -18, the value of R is a minimum. Thus, the number of units that maximize the revenue = 18 units

i. The maximum revenue is given as

            R = 14 + 81 + 18 - ((18)3)/12

                        = Shs. 986

ii.The price per unit to maximize the revenue is given as:

986/18 = 54.78 or Shs.54.78

Posted Date: 2/15/2013 6:51:53 AM | Location : United States







Related Discussions:- Example of inflection point-differential equation, Assignment Help, Ask Question on Example of inflection point-differential equation, Get Answer, Expert's Help, Example of inflection point-differential equation Discussions

Write discussion on Example of inflection point-differential equation
Your posts are moderated
Related Questions

Program of "surface of revolution" in MATLAB

Common Graphs : In this section we introduce common graph of many of the basic functions. They all are given below as a form of example Example   Graph y = - 2/5 x + 3 .

log4^(x+2)=log4^8

Problem 1 Let ~x0 = A~x and y 0 = B~y be two 2  2 linear systems of ODE. (1) Suppose that A and B have the same purely imaginary eigenvalues. Prove that these systems are topologi


Prisoners Dilemma This is a type of non-zero sum game and derives its name from the given story: The district attorney has two bank robbers in separate cells and offers them

Samantha owns a rectangular field that has an area of 3,280 square feet. The length of the field is 2 more than twice the width. What is the width of the field? Let w = the wid

Owner of a computer repair shop has daily revenue with mean $7200 and SD $1200 Daily revenue for next 30 days will be monitored. What is probability that daily revenue for those 30

Describe the System with 3 Variables ? This is an example of solving a system of equations using the substitution method. Warning: You will not understand this example if you