Applications of Derivatives, Method of Finding Maxima or Minima, Use of Second Derivative for Determination of Maximum and Minimum Values of a Function

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Method of Finding Maxima or Minima

We have arrived at the method of finding the maxima or minima of a function. It is as follows:

  1. Find

  2. Put and find stationary points

  3. Consider the sign of in the neighbourhood of stationary points. If it changes sign from to , then has maximum value at that point and if changes sign from to , then has minimum value at that point.

  4. If does not change sign in the neighbourhood of a point then it is a point of inflexion.

Example:

Find the maximum (local maximum) and minimum (local minimum) points of the function .

Solution:

Here

Step I:

gives us

Or

Or

Or

Stationary points are

Step II:

For and

For

changes sign from to in the neighbourhood of .

has minimum value at .

Step III:

For and

For

changes sign from to in the neighbourhood of .

has maximum value at .

and give us points of maxima and minima respectively. If we want to find maximum value (minimum value), then we have

maximum value

And minimum value

and are points of local maxima and local minima respectively.

Use of Second Derivative for Determination of Maximum and Minimum Values of a Function:

We now give below another method of finding local maximum or minimum of a function whose second derivative exists. Various steps used are:

  1. Let the given function be denoted by .

  2. Find and equate it to zero.

  3. Solve , let one of its real roots be .

  4. Find its second derivative,. For every real value ‘’ of obtained in step (3), evaluate. Then if

then is a point of local maximum.

then is a point of local minimum.

then we use the sign of on the left of ‘’ and on the right of ‘’ to arrive at the result.

Example:

Find the local minimum of the following function:

Solution:

Let

Then

For local maximum or minimum

or

Now

For

is a point of local maximum.

And is a local maximum.

For

is a point of local minimum

And is a local minimum.

Applications of Maxima and Minima to Practical Problems:

The application of derivative is a powerful tool for solving problems that call for minimising or maximising a function. In order to solve such problems, we follow the steps in the following order:

  1. Frame the function in terms of variables discussed in the data.

  2. With the help of the given conditions, express the function in terms of a single variable.

  3. Lastly, apply conditions of maxima or minima as discussed earlier.

Example:

Find two positive real numbers whose sum is and their product is maximum.

Solution:

Let one number be . As their sum is , the other number is . As the two numbers are positive, we have,

Let their product be

Then

We have to maximize the product .

We, therefore, find and put that equal to zero.

For maximum product,

Or

Or

Now which is negative. Hence is maximum at

The other number is

Hence the required numbers are .

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