Applications of Derivatives, Langrange's Mean Value Theorem, Mathematical Formulation of the Theorem, Increasing and Decreasing Functions, Monotonic Functions

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Langrange’S Mean Value Theorem

  • This theorem improves the result of Rolle’s Theorem saying that it is not necessary that tangent may be parallel to x-axis. This theorem says that the tangent is parallel to the line joining the end points of the curve.

  • In other words, this theorem says that there always exists a point on the graph, where the tangent is parallel to the line joining the end-points of the graph.

Mathematical Formulation of the Theorem

Let be a real valued function defined on the closed interval such that

(a) is continuous on , and

(b) is differentiable in

(c)

Then there exists a point in the open interval such that

Example:

Verify Langrange’s Mean value theorem for on

Solution:

Or

is a polynomial function and hence continuous and differentiable in the given interval

Here,

All the conditions of Mean value Theorem are satisfied

Now

or

or

Langrange’s mean value theorem is verified

Increasing and Decreasing Functions

  • You have already seen the common trends of an increasing or a decreasing function. Here we will try to establish the condition for a function to be an increasing or a decreasing.

  • Let a function be defined over the closed interval .

  • Let , then the function is said to be an increasing function in the given interval if whenever . It is said to be strictly increasing if for all , .

  • In Fig., increases from to as increases from to .

Graph of increases of sin x

Graph of Increases of Sin X

Graph of increases of sin x

In Fig., decreases from to as increases from to .

Graph of decreases of cos x

Graph of Decreases of Cos X

Graph of decreases of cos x

Monotonic Functions

  • Let be any two points such that in the interval of definition of a function .

  • Then a function is said to be monotonic if it is either increasing or decreasing. It is said to be monotonically increasing if for all belonging to the interval and monotonically decreasing if .

Example:

Prove that the function is monotonic for all values of .

Solution:

Consider two values of say

Such that .....

Multiplying both sides of by , we have .....

Adding to both sides of , to get

We have

Thus, we find whenever .

Hence the given function is monotonic function. (Monotonically increasing).

Theorem 1: If is an increasing function on an open interval ]a, b[, then its derivative

is positive at this point for all .

Theorem 2: If is a decreasing function on an open interval ]a, b[ then its derivative is negative at that point for all .

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