Maxima and minima, Mathematics

Maxima and Minima

We have to make a distinction between relative maxima (or minima) and global maxima (or minima). Let f(x) be a function of x. Then the global maxima of f(x) is the maximum value which f(x) could take for all the possible range of values of x. For example, if x denotes the number of units produced and f(x) denotes the total profit when x units are produced, then we are interested in the maximum value of f(x) for x in the range 0 to the maximum production capacity. Compared to the global maximum, we speak of relative maxima, which is the maximum value of f(x) in a relatively small range of values of x. f(x) is said to have a relative maxima at a point x = a, if the value of the function at a, f(a) is greater than or equal to f(x) for values of x sufficiently close to value a. Look at the graph of a function f(x) given below.

Figure 

10_maxima and minima.png

This function has a relative maxima at points x = a and at x = c, and a relative minima at points x = b and x = d. If we consider values of x in the range 0 to d, then the global maxima of the function is f(c) attained at point x = c.

The following characteristic features of the relative maxima and minima could be observed:

  1. For values of x which are sufficiently close to a(or c), the value of the function first increases and then decreases. The slope of the function is positive up to a(or c) and then it becomes negative. For points b and d, the slope changes from negative to positive.

  2. The tangent lines at points a, b, c and d are horizontal indicating that the slopes at those points are zero.

The above observations give the following results:

Result 1:    If f(x) has a relative maxima or minima at  x = a, then the value of the derivative of f(x) at a,  138_maxima and minima14.png is 0.

Note that the converse may not be true. If at a point a,  138_maxima and minima14.png  is 0, it need not necessarily imply that f(x) has a relative maxima or minima.

Result 2:    If, at point a, we have  138_maxima and minima14.png  = 0 and  138_maxima and minima14.png  < 0 then f(x) has a relative maxima at x = a.

If  138_maxima and minima14.png = 0 and  138_maxima and minima14.png  > 0 then f(x) has a relative minima at x = a.

This is because at relative maxima (or minima) the slope changes from positive (or negative) to negative (or positive) for points of close to a.

If  138_maxima and minima14.png = 0 and  138_maxima and minima14.png  = 0 then we cannot say anything definitely about the occurrence of maxima or minima at point a.

Example 

A manufacturer fixed the following demand curve:

         P(x)   = 200 - 3x

Where x = quantity and P = price

Let the fixed cost of production be Rs.25 and the variable cost Rs.2 per unit. The manufacturer wants to fix the level of output so as to maximize his profit.

The profit function f(x) is:

f(x)    = total revenue - total cost

         = price x quantity - (fixed cost + variable cost)

         = (200 - 3x)x - (25 + 2x)

         = -3x2 + 198x - 25

The first order derivative is:

f'x

= - 6x + 198

 

This is 0 if - 6x + 198 = 0 or x = 198/6 = 33

At this level of output, i.e. x = 33, we would like to examine whether we have a minimum or maximum. The second order derivative is:

 

f''(x)

= - 6 which is negative for all values of x

 

Therefore, 138_maxima and minima14.png  < 0

Hence, we conclude that at the level of production of 33 units, the manufacturer will make a maximum profit. The maximum profit is:

-3  *  (33)2 + 198  *  33 - 25 = Rs.3,242

Suppose, the quantity that could be produced is limited to the range, say, 10 to 30 units, then we have to check the value of the profit at the boundary points 10 and 30, in addition to any other points in the range (10,30) and then find the optimal level of production.

Posted Date: 9/13/2012 7:21:16 AM | Location : United States







Related Discussions:- Maxima and minima, Assignment Help, Ask Question on Maxima and minima, Get Answer, Expert's Help, Maxima and minima Discussions

Write discussion on Maxima and minima
Your posts are moderated
Related Questions
Find the 20 th term from the end of the AP 3, 8, 13........253. Ans:    3, 8, 13 .............. 253 Last term = 253 a20 from end = l - (n-1)d 253 - ( 20-1) 5 253


Total Contribution per Year for next 10yeras =$1000+$800 =$1800 So Total Future fund Vaule  =$1800*(1+1.073+power(1.073,2)+ power(1.073,2)+ power(1.073,3)+ power(1.073,4)+ power

The two sides of a triangle are 17 cm and 28 cm long, and the length of the median drawn to the third side is equal to 19.5 cm. Find the distance from an endpoint of this median to

what is 24 diveded by 3

RATIONAL NUMBERS All numbers of the type p/q where p and q are integer and q ≠0, are known as rational. Thus  it can be noticed that every integer is a rational number

A bank pays on its savings an interest rate of 6% per year but compounds interest monthly (i.e., estimates the interest each month and adds it to the balance).  You plan to deposit

assignment on income tax


Case 1: Suppose we are given expressions like 3abc and 7abc and asked to compute their sum. If this is the case we should not worry much. Because adding like exp