Applications of Derivatives, Maximum and Minimum Values of a Functions, Conditions for Maximum or Minimum

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Maximum and Minimum Values of a Functions

  • We have seen the graph of a continuous function. It increases and decreases alternatively. If the value of a continuous function increases upto a certain point then begins to decrease, then this point is called point of maximum and corresponding value at that point is called maximum value of the function.

  • A stage comes when it again changes from decreasing to increasing. If the value of a continuous function decreases to a certain point and then begins to increase, then value at that point is called minimum value of the function and the point is called point of minimum.

Graph of Maximum and Minimum Values of a Functions

Graph of Maximum and Minimum Values of a Functions

Graph of Maximum and Minimum Values of a Functions

Above Fig. shows that a function may have more than one maximum or minimum values. So, for continuous function we have maximum (minimum) value in an interval and these values are not absolute maximum (minimum) of the function. For this reason, we sometimes call them as local maxima or local minima.

A function f (x) is said to have a maximum or a local maximum at the point where a

(See Fig. (a)), if for all sufficiently small positive .

A maximum (or local maximum) value of a function

A Maximum (Or Local Maximum) Value of a Function

A maximum (or local maximum) value of a function

  • A maximum (or local maximum) value of a function is the one which is greater than all other values on either side of the point in the immediate neighbourhood of the point.

  • A function is said to have a minimum (or local minimum) at the point if

  • where for all sufficiently small positive .

  • In Fig. (b), the function has local minimum at the point .

  • A minimum (or local minimum) value of a function is the one which is less than all other values, on either side of the point in the immediate neighbourhood of the point.

Conditions for Maximum or Minimum:

Graph of Maximum and Minimum Values of a Functions

Graph of Maximum and Minimum Values of a Functions

Graph of Maximum and Minimum Values of a Functions

We know that derivative of a function is positive when the function is increasing and the derivative is negative when the function is decreasing. We shall apply this result to find the condition for maximum or a function to have a minimum. Refer to Fig., points are points of maxima and points are points of minima.

Now, on the left of , the function is increasing and so , but on the right of , the function is decreasing and, therefore,. This can be achieved only when becomes zero somewhere in between. We can rewrite this as follows:

A function has a maximum value at a point if (i) and

(ii) changes sign from positive to negative in the neighbourhood of the point at which (points taken from left to right).

Now, on the left of (See Fig.), function is decreasing and therefore, is negative and on the right of , is increasing and so is positive. Once again will be zero before having positive values. We rewrite this as follows:

A function has a minimum value at a point if (i) , and (ii) changes sign from negative to positive in the neighbourhood of the point at which .

We should note here that is necessary condition and is not a sufficient condition for maxima or minima to exist. We can have a function which is increasing, then constant and then again increasing function. In this case, does not change sign. The value for which is not a point of maxima or minima. Such point is called point of inflexion.

Stationary point of Function

Stationary Point of Function

Stationary point of Function

  • For example, for the function is the point of inflexion as does not change sign as passes through . is positive on both sides of the value (tangents make acute angles with x-axis) (See Fig.).

  • Hence has a point of inflexion at .

  • The points where are called stationary points as the rate of change of the function is zero there. Thus points of maxima and minima are stationary points.

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