Quick Facts
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Monotonicity (Increasing and Decreasing Function) is considered one of the most asked concept.
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52 Questions around this concept.
Solve by difficulty
MEDIUMLet be a function defined by
, and
.
Consider two statements
(I) is an increasing function in
(II) is one-one in
Then,
Let f : [2, 4] be a differentiable function such that
with f
=
and f(4) =
Consider the following two statements
The set of all for which the equation
has exactly one real root, is
The value of , where [.] denotes greatest integer function, is
Let $\mathrm{g}(\mathrm{x})=3 \mathrm{f}\left(\frac{\mathrm{x}}{3}\right)+\mathrm{f}(3-\mathrm{x})$ and $\mathrm{f}^{\prime \prime}(\mathrm{x})>0$ for all $\mathrm{x} \in(0,3)$. If $\mathrm{g}$ is decreasing in $(0, \alpha)$ and increasing in $(\alpha, 3)$, then $8 \alpha$ is:
Consider the function $\mathrm{f}:\left[\frac{1}{2}, 1\right] \rightarrow \mathbb{R}$ defined by $\mathrm{f}(\mathrm{x})=4 \sqrt{2} \mathrm{x}^3-3 \sqrt{2} \mathrm{x}-1$.
Consider the statements.
(I) The curve $y=f(x)$ intersects the $x$-axis exactly at one point.
(II) The curve $y=f(x)$ intersects the $x$-axis at $x=\cos \frac{\pi}{12}$.
Then
The function $f(x)=\frac{x}{x^2-6 x-16}, x \in \mathbb{R}-\{-2,8\}$
Let $f: R \rightarrow(0, \infty)$ be strictly increasing function such that $\lim _{x \rightarrow \infty} \frac{f(7 x)}{f(x)}=1$. Then, the value of $\lim _{x \rightarrow \infty}\left[\frac{f(5 x)}{f(x)}-1\right]$ is equal to
The function in its entire domain is:
Consider the function $\mathrm{f}(0,2) \rightarrow \mathrm{R}$ defined by $\mathrm{f}(\mathrm{x})=\frac{\mathrm{x}}{2}+\frac{2}{\mathrm{x}}$ and the function $\mathrm{g}(\mathrm{x})$ defined by
$\mathrm{g}(\mathrm{x})=\left\{\begin{array}{ll}\min \{\mathrm{f}(\mathrm{t})\}, & 0<\mathrm{t} \leq \mathrm{x} \text { and } 0<\mathrm{x} \leq 1 \\ \frac{3}{2}+\mathrm{x}, & 1<\mathrm{x}<2\end{array}\right.$. Then,
Concepts Covered - 4
Monotonicity (Increasing and Decreasing Function)
A function is said to be monotonic if it is either increasing or decreasing in its entire domain.
Increasing Function
A function f(x) is increasing in [a, b] if f(x2) ≥ f(x1) for all x2 > x1, where x1, x2 ∈ [a, b].
If a function is differentiable, then
A function is said to be increasing if it is increasing in its entire domain.
Example:
- f(x) = x is increasing in R. (As f '(x) = 1, so f '(x) ≥ 0 for all values of x in R, so it is an increasing in R).
- f(x) = tan-1x, is also an increasing function on R as f '(x) ≥ 0 for all real values of x.
- f(x) = [x] is also an increasing function on R. Its differentiation is not defined at all points, but from its graph we can see that on giving higher value of x to this function, it returns equal or higher value of y. For this function x2 > x1 implies f(x2) ≥ f(x1). Hence it is an increasing function.
Note:
These functions are also simply called 'increasing functions' as they are increasing in their entire domains.
f(x) = ln(x) is increasing function as it is increasing in its entire domain but it is not increasing in R (as it is not defined for x < 0 and x = 0)
So tangent to the curve, f(x) at each point makes an acute angle with a positive direction of x-axis or parallel to the x-axis.
Strictly Increasing Function
A function f(x) is strictly increasing in interval [a,b] if f(x2) > f(x1) for all x2 > x1, where x1, x2 ∈ [a,b].
If a function is differentiable, then
So tangent to the curve, f(x) at each point makes an acute angle with the positive direction of the x-axis.
Example: f(x) = x is strictly increasing but f(x) = [x] is not strictly increasing
Note:
If f '(x) = 0 at some discrete points (if number of such points can be counted), and at other points f '(x) > 0, still the function is strictly increasing function.
Example
Consider f(x) = [ x ], where [ . ] is the greatest integer function.
For this function x2 > x1 does not always implies f(x2) > f(x1)
However, x2 > x1 does imply f(x2) ≥ f(x1)
So, f(x) = [ x ] is increasing function but not a strictly increasing function.
Let’s look into some more examples,
Functions ex , ax (a > 1), x3 + x are strictly increasing functions in their entire domain.
Concavity
When you draw a tangent at any point on the curve, if the entire curve lies above the tangent, in this case, the curve is called a concave upward curve.
And if the entire curve lies below the tangent then the curve is called a concave downward curve.
Strictly Increasing functions can be classified as:
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Concave up: When f’(x) > 0 and f”(x) > 0 ∀ x ∈ domain
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Concave down: When f’(x) > 0 and f“(x) < 0 ∀ x ∈ domain
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When f’(x) > 0 and f”(x) = 0 ∀ x ∈ domain
Decreasing Function
A function f(x) is decreasing in the interval [a,b] if f(x2) ≤ f(x1) for all x2 > x1, where x1, x2 ∈ [a,b]
If a function is differentiable, then
Example
- f(x) = - x is decreasing in R (As f '(x) = -1, so f '(x) < 0 for all real values of x. We can also see that it is decreasing from its graph)
- f(x) = e-x is decreasing in R (As f '(x) = -e-x, so f '(x) < 0 for all real values of x. We can also see that it is decreasing from its graph)
- f(x) = cot(x) is decreasing in
- f(x) = cot-1(x) is decreasing in R
A function is said to be decreasing if it is decreasing in its entire domain.
So tangent to the curve, f(x) at each point makes an obtuse angle with the positive direction of x-axis or parallel to the x-axis.
Strictly Decreasing Function
A function f(x) is strictly decreasing in its domain (Df) if f(x2) < f(x1) for all x2 > x1, where x1, x2 ∈ Df.If a function is differentiable in domain (Df) then
So tangent to the curve, f(x) at each point makes an obtuse angle with positive direction of x-axis.
For example, functions e-x and -x3 are strictly decreasing functions.
Note:
If f '(x) = 0 at some discrete points (if number of such points can be counted), and at other points f '(x) > 0, still the function is strictly increasing function.
NOTE:
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If a function is not differentiable at all points, this does not mean that function is not increasing or decreasing. A function may increase or decrease on an interval without having a derivative defined at all points.
For example, y = x1/3 is increasing everywhere including x = 0, but the derivative is not defined at this point as the function has vertical tangent.
Decreasing functions can be classified as:
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Concave up: When f’(x) < 0 and f”(x) > 0 ∀ x ∈ domain
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Concave down: When f’(x) < 0 and f“(x) < 0 ∀ x ∈ domain
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When f’(x) > 0 and f”(x) = 0 ∀ x ∈ domain
Monotonicity of Composite Function
The nature of monotonicity of composite function f(g(x)) and g(f(x)) depends on the nature of the function f(x) and g(x).
If f(x) is increasing function and g(x) is decreasing function, then for x2 > x1, we have f(x2) ≥ f(x1) and g(x2) ≤ g(x1).
So, for x2 > x1, we have f(g(x2)) ≤ f(g(x1)) and g(f(x2)) ≤ g(f(x1)) .
Thus, f(g(x)) is a decreasing function and also, g(f(x)) is also a decreasing function.
If both f(x) and g(x) are increasing or decreasing function, then f(g(x)) and g(f(x)), i.e., both composite functions are increasing.
For differentiable functions, we can prove it in another way
If f(x) and g(x) are differentiable function, with f(x) increasing and g(x) decreasing, then
Similarly, all the possibilities of the nature of the composite function f(g(x)) and g(f(x)) are given below
AID TO MEMORY:
Where (+) means strictly increasing and (-) means strictly decreasing.
Non-Monotonic Function and Critical Point
A function which are neither always increasing nor always decreasing in their domain are called non-monotonic functions.
For example,
f(x) = sin x, which is increasing in the first quadrant and the fourth quadrant and decreasing in second and third quadrant.
Consider another function, y = f(x) = |x2 - 2|
f(x) is increases in [-√2, 0] and [√2, ∞ ) and decreases in (-∞, -√2] and [0,√2]
Hence this function is a non-monotonic function.
Critical Points
A critical point of a function is a point where its derivative does not exist or its derivative is equal to zero.
All the values of ‘x’ obtained by below conditions are said to be the critical points.
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f(x) does not exists
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f’(x) does not exists
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f’(x) = 0
Critical points are interior points of the intervals.
For the function f(x) = | x2 - 4|, critical points are x = +2, -2 and x = 0 where its derivative is zero.
Study it with Videos
Monotonicity (Increasing and Decreasing Function)
Decreasing Function
Monotonicity of Composite Function
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