5.4 Monotonic functions  (Page 2/3)

Derivative and nature of function

We shall learn subsequently that first derivative of a function is defined in terms of ratio of the differences between two successive values of function and that of independent variable, however small is the difference in independent variable. It is easy to visualize, then, that if we know the nature of derivative in a given interval, then we can determine monotonic behavior of the function as well.

$$f\prime \left(x\right)=\underset{h\to 0}{\overset{}{Lim}}\frac{f\left(x+h\right)-f\left(x\right)}{h}$$

Depending on the monotonic nature of function, the relative values of f(h) and f(x+h) are different and so the sign of first derivative.

Strictly increasing function

The successive value of function increases as the value of the independent variable increases. In other words, the preceding values are less than successive values that follow. Mathematically,

$$\text{If}{x}_1<x_{\mathrm{2,}}\text{then}f\left(x_1\right)<f\left(x_2\right)$$

As f( $x_1$ )<f( $x_2$ ) for all $x_1$ , $x_2$ ∈X, the difference “f(x+h) – f(x)” is positive for “h”, however small. This implies that the first derivative of function is positive. If we think of possibility, then we can realize that tangent to the function curve can be parallel to x-axis for couple of x values, while curve is continuously increasing in the interval. It means that first derivative can be equal to zero for few points in the interval in which it is strictly increasing. This is clear from the figure given here,

Thus, for strictly increasing function,

$$f\prime \left(x\right)\ge 0;\text{Equality sign holds for points only - not on a continuous section in X}$$

For strictly increasing function, if $x_1$ < $x_2$ , then f( $x_1$ )<f( $x_2$ ), for all $x_1$ , $x_2$ ∈X. It means that all distinct x values correspond to distinct y values and vice-versa. Therefore, strictly increasing function is one-one function i.e. a bijection and hence “invertible”. In other words, if a function has strict increasing order, then it is invertible. Mathematically, we say that if f’(x) ≥0; (equality holding for points only), x∈X, then function is invertible in X. For example, consider sine function,

$$f\left(x\right)=\sin x$$ $$\Rightarrow f\prime \left(x\right)=\cos x$$

We know that cosx is positive in the interval [-π/2,π/2]. Hence sine function is a strictly increasing function in [-π/2,π/2]and is invertible. Recall that inverse sine function is defined in this interval.

The order of a function provides an easy technique to determine range of a continuous function, corresponding to a given domain interval. For example, if domain of a continuously increasing function, f(x), is [ $x_1$ , $x_2$ ], then the least value of the function is f( $x_1$ ) and greatest value of the function is f( $x_2$ ). Hence, range of the function is :