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| Increasing and Decreasing Functions |
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| This section explains how derivative can be used to check whether a function is increasing, decreasing or neither increasing nor decreasing in its domain. |
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| Let f be a function defined on an interval I and let x1 and x2 be any two points on I. |
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| (i) f is said to be increasing in the interval I, |
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| Example: |
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| Define |
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| The graph of the function is as follows: |
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| f(x) is increasing, because |
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| Since 2 - 1 < 3 - 1 |
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| (ii) f is said to strictly increasing in the interval I if |
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| For example, |
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| Let x1 < x2 |
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| f(x) is strictly increasing function. |
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(iii) f(x) is said to be decreasing function if for x1, x2  I |
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| For example, |
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| f(x) = 1 - x for 0 < x < 1 |
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= 0 for 0  x <2 |
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= 2 - x for x  2 |
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| is a decreasing function. |
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| (iv) f (x) is said to be strictly decreasing on an interval I if for |
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| For example, |
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 is strictly decreasing function. |
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| Let f be continuous on [a, b] and differentiable on the open interval (a, b). Then |
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(a) f is increasing on [a, b] if f '(x) > 0 for each x  (a, b) |
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(b) f is decreasing on [a, b] if f '(x) < 0 for each x  (a, b) |
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| This theorem can be proved by using Mean Value Theorem. We shall prove the theorem after learning Mean Value Theorem. |
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| This theorem is applied in various problems to check whether a function is increasing or decreasing. |
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| (1) Let the given function be f (x) on the real number line R. |
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| (2) Differentiate the function f(x) with respect to x and equate it to zero i.e., put f '(x) = 0. Solve for x. These values of x which satisfy f '(x) = 0 are called Critical values of the function |
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| (3) Arrange these Critical values in ascending order and partition the domain of f (x) into various intervals, using the Critical values. |
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| (4) Check the sign of f '(x) in each open intervals. |
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| (5) If f '(x) > 0 in a particular interval, then the function is increasing in that particular interval. |
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| If f '(x) < 0 in a particular interval, then the function is decreasing in that particular interval. |
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| Example: |
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| Find the intervals on which the function |
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| (a) increasing (b) decreasing |
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| Differentiating the function, we have |
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The critical values in ascending order are -1, 1. We divide the Real numbers into the intervals  |
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| = - ve |
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Since f '(x) < 0, the function is decreasing in the interval  . |
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| Since f '(x) > 0 in the interval (-1, 1), the function is increasing in this interval. |
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| = - ve |
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Since f '(x) < 0, f(x) is decreasing in the interval (1,  ). |
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