# Maxima and minima properties problem

• zorro
In summary, the conversation discusses finding the value of the common difference (c.d.) in an arithmetic progression (A.P.) that would make the product of three terms (A3, A7, A12) the greatest. The attempt at a solution involves using the AM-GM inequality and the properties of maxima and minima to find the value of d. The final answer is not 18, as previously thought, but can be obtained by modifying the approach.
zorro

## Homework Statement

Let An be the nth term of an A.P. and if A7 = 15, then the value of the c.d. that would make A3 x A7 x A12 greatest is :

1)9
2)9/4
3)3/8
4)18

## The Attempt at a Solution

Applying AM>=GM

A3+A7+A12/3 >= (A3 x A7 x A12)^1/3

given that A+6d=15
therefore 3A+19d=45

The previous expression reduces to 45+d/3 >= (A3 x A7 x A12)^1/3
cubing the inequality.
(45+d/3)^3 >= (A3 x A7 x A12)

From the choices, 18 will make RHS greatest. But that is not correct!
Any help appreciated.

nice one.. it is.. sure some iit problem
I believe your 2nd last statement is wrong because 18 makes the LHS greatest not RHS and RHS has to be lesser than LHS.

Actually i tried a different approach and i'll give you a hint.
Use the maxima and minima properties and differentiate finding out the values of d. (you will get 2 values ..though one will be eliminated)

Yes, 18 makes the LHS greatest. Since RHS is </= LHS, greater LHS implies greater RHS.
I just wanted to know what was wrong in this.

Your method doesn't exactly give you the answer... it just tells you when LHS is greatest and gives no info about RHS(which is what you want). Though this approach may be modified to get the answer.
(I will try that out)

## What is a "Maxima and minima properties problem"?

A "Maxima and minima properties problem" is a type of problem in mathematics that involves finding the maximum and minimum values of a given function over a specific interval or domain. This is often done by finding the critical points of the function and evaluating the function at those points.

## What is the significance of finding maxima and minima in a function?

Finding the maxima and minima of a function can provide valuable information about the behavior and characteristics of the function. It can help determine the optimal values for a given situation, such as the maximum profit or minimum cost, and can also help identify important points on a graph, such as points of intersection or points of inflection.

## How do you find the maxima and minima of a function?

To find the maxima and minima of a function, you first need to find the critical points, which are the points where the derivative of the function is equal to zero or undefined. Then, you can use the first or second derivative test to determine whether these critical points are local maxima, local minima, or neither.

## What is the difference between global and local maxima and minima?

A global maximum or minimum is the largest or smallest value of a function over its entire domain, while a local maximum or minimum is the largest or smallest value of a function within a specific interval or neighborhood. In other words, a global maximum or minimum is the absolute maximum or minimum of a function, while a local maximum or minimum is a relative maximum or minimum within a smaller range.

## What are some real-world applications of maxima and minima problems?

Maxima and minima problems have numerous applications in various fields, such as economics, engineering, physics, and chemistry. They can be used to optimize production processes, determine the most efficient use of resources, analyze market trends, and solve optimization problems in science and technology.

Replies
6
Views
1K
Replies
4
Views
4K
Replies
7
Views
2K
Replies
10
Views
3K
Replies
8
Views
3K
Replies
2
Views
1K
Replies
3
Views
2K
Replies
1
Views
2K
Replies
5
Views
2K
Replies
3
Views
4K