Monotonicity Formula Sheet

Monotonicity refers to the property of a function where it consistently either increases or decreases over its entire domain, without changing direction.

Neetesh Kumar | June 02, 2024 $\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space$ Share this Page on:

• 1. Nature of a Function at a Point
• 2. Increasing & Decreasing Nature of a Function Over an Interval
• 3. Rolle's Theorem
• 4. Lagrange's Mean Value Theorem (LMVT)
• 5. Taylor's Theorem
• 6. Maclaurin's Theorem
• 7. Special Points to Note

1. Nature of a Function at a Point:

(a). Increasing at $x = a$:

If $f(s) < f(a) < f(t)$ whenever $s < a < t$, where $s, t \in (a - h, a + h) \cap D_f$ for some $h > 0$, then $f$ is said to be increasing at $x = a$.

• When $a$ is the left end of the interval: $f(a) < f(x) \quad \forall x \in (a, a + h) \cap D_f \text{ for some } h > 0$ $\Rightarrow f$ is increasing at $x = a$.

• When $a$ is the right end of the interval: $f(x) < f(a) \quad \forall x \in (a - h, a) \cap D_f \text{ for some } h > 0$ $\Rightarrow f$ is increasing at $x = a$.

(b). Strictly Increasing at $x = a$:

If $f(s) < f(a) < f(t)$ whenever $s < a < t$, where $s, t \in (a - h, a + h) \cap D_f$ for some $h > 0$, then $f$ is said to be strictly increasing at $x = a$.

• When $a$ is the left end of the interval: $f(a) < f(x) \quad \forall x \in (a, a + h) \cap D_f \text{ for some } h > 0$ $\Rightarrow f$ is strictly increasing at $x = a$.

• When $a$ is the right end of the interval: $f(x) < f(a) \quad \forall x \in (a - h, a) \cap D_f \text{ for some } h > 0$ $\Rightarrow f$ is strictly increasing at $x = a$.

(c). Decreasing at $x = a$:

If $f(s) \geq f(a) \geq f(t)$ whenever $s < a < t$, where $s, t \in (a - h, a + h) \cap D_f$ for some $h > 0$, then $f$ is said to be decreasing at $x = a$.

• When $a$ is the left end of the interval: $f(a) \geq f(x) \quad \forall x \in (a, a + h) \cap D_f \text{ for some } h > 0$ $\Rightarrow f$ is decreasing at $x = a$.

• When $a$ is the right end of the interval: $f(x) \geq f(a) \quad \forall x \in (a - h, a) \cap D_f \text{ for some } h > 0$ $\Rightarrow f$ is decreasing at $x = a$.

(d). Strictly Decreasing at $x = a$:

If $f(s) > f(a) > f(t)$ whenever $s < a < t$, where $s, t \in (a - h, a + h) \cap D_f$ for some $h > 0$, then $f$ is said to be strictly decreasing at $x = a$.

• When $a$ is the left end of the interval: $f(a) > f(x) \quad \forall x \in (a, a + h) \cap D_f \text{ for some } h > 0$ $\Rightarrow f$ is strictly decreasing at $x = a$.

• When $a$ is the right end of the interval: $f(x) > f(a) \quad \forall x \in (a - h, a) \cap D_f \text{ for some } h > 0$ $\Rightarrow f$ is strictly decreasing at $x = a$.

2. Increasing & Decreasing Nature of a Function Over an Interval:

Consider an interval $I \subseteq D_f$

(a). Increasing Over an Interval $I$:

If $\forall x_1, x_2 \in I$, $x_1 < x_2 \Rightarrow f(x_1) < f(x_2)$, then $f$ is increasing over the interval $I$.

(b). Decreasing Over an Interval $I$:

If $\forall x_1, x_2 \in I$, $x_1 < x_2 \Rightarrow f(x_1) > f(x_2)$, then $f$ is decreasing over the interval $I$.

(c). Strictly Increasing Over an Interval $I$:

If $\forall x_1, x_2 \in I$, $x_1 < x_2 \Leftrightarrow f(x_1) < f(x_2)$, then $f$ is strictly increasing over the interval $I$.

(d). Strictly Decreasing Over an Interval $I$:

If $\forall x_1, x_2 \in I$, $x_1 < x_2 \Leftrightarrow f(x_1) > f(x_2)$, then $f$ is strictly decreasing over the interval $I$.

Monotonic Function:

If a function is either increasing or decreasing over an interval, then it is said to be a monotonic function over the interval.
If a function is either strictly increasing or strictly decreasing over an interval, then it is said to be a strictly monotonic function over the interval.

Some Important Points:

1. Increasing or monotonic increasing or non-decreasing has the same meaning. Similarly, decreasing or monotonic decreasing or non-increasing has the same meaning.
2. If a function is strictly increasing, it is also said to be an increasing function, but the converse is not necessarily true.
3. Functions that are increasing over some interval and decreasing over another interval are known as non-monotonic functions over the union of the intervals.
4. A function may be monotonic in a subset but not in a superset.
5. Constant function is increasing as well as decreasing over any interval. So it is called a monotonic function.

For Differentiable Functions:

Consider an interval $I (\subseteq D_f)$ that can be $[a, b]$ or $(a, b)$ or $[a, b)$ or $(a, b)$.

1. $f'(x) > 0 \quad \forall x \in I \Rightarrow f$ is a strictly increasing function over the interval $I$.
2. $f'(x) \geq 0 \quad \forall x \in I \Rightarrow f$ is an increasing function over the interval $I$.
3. $f'(x) > 0 \quad \forall x \in I$ and $f'(x) = 0$ do not form any interval (that means $f'(x) = 0$ at discrete points)
   $\Rightarrow f \text{ is a strictly increasing function over the interval } I$
4. $f'(x) < 0 \quad \forall x \in I \Rightarrow f$ is a strictly decreasing function over the interval $I$.
5. $f'(x) \leq 0 \quad \forall x \in I \Rightarrow f$ is a decreasing function over the interval $I$.
6. $f'(x) < 0 \quad \forall x \in I$ and $f'(x) = 0$ do not form any interval (that means $f'(x) = 0$ at discrete points)
   $\Rightarrow f \text{ is a strictly decreasing function over the interval } I$

3. Rolle's Theorem:

Let $f$ be a function that satisfies the following three hypotheses:

1. $f$ is continuous in the closed interval $[a, b]$.
2. $f$ is differentiable in the open interval $(a, b)$.
3. $f(a) = f(b)$.

Then there is a number $c$ in $(a, b)$ such that $f'(c) = 0$.

If $f$ is a differentiable function, then between any two consecutive roots of $f(x) = 0$, there is at least one root of the equation $f'(x) = 0$.

4. Lagrange's Mean Value Theorem (LMVT):

Let $f$ be a function that satisfies the following hypotheses:

1. $f$ is continuous in a closed interval $[a, b]$.
2. $f$ is differentiable in the open interval $(a, b)$.

Then there is a number $c$ in $(a, b)$ such that: $f'(c) = \frac{f(b) - f(a)}{b - a}$

(a) Geometrical Interpretation:

Geometrically, the Mean Value Theorem says that somewhere between $A$ and $B$, the curve has a tangent parallel to the secant $AB$.

(b) Physical Interpretation:

If we think of the number $\frac{f(b) – f(a)}{b – a}$ as the average change in f over [a, b] and f'(c) as an instantaneous change, then the Mean Value Theorem says that at some interior point the instantaneous change must equal the average change over the entire interval.

Special Cases:

1. If $f(a) = f(b)$, then $f'(c) = 0$ for some $c \in (a, b)$ (this is Rolle's Theorem).
2. If $f'(x) = 0 \quad \forall x \in (a, b) \Rightarrow f$ is constant on $(a, b)$.
3. Let $f'(x) = g'(x) \quad \forall x \in (a, b) \Rightarrow f - g$ is constant on $(a, b)$.
4. Let $f'(x) = 0 \quad \forall x \in (a, b)$ and $f$ is continuous at $x = a \Rightarrow f$ is constant on $[a, b]$.

5. Taylor's Theorem:

Let $f$ be a function that is $(n + 1)$ times continuously differentiable on an interval $I$ containing $a$. Then for each $x$ in $I$:

$f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x - a)^n + R_n(x)$

where the remainder term $R_n(x)$ is given by:

$R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x - a)^{n+1}$ for some $c$ in the interval between $a$ and $x$.

6. Maclaurin's Theorem:

Maclaurin's Theorem is a special case of Taylor's Theorem when $a = 0$. It states that if $f$ is $(n + 1)$ times continuously differentiable at $0$, then:

$f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \cdots + \frac{f^{(n)}(0)}{n!}x^n + R_n(x)$

where the remainder term $R_n(x)$ is given by:

$R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}x^{n+1}$ for some $c$ in the interval between $0$ and $x$.

7. Special Points to Note:

One can use monotonicity to identify the number of roots of the equation in a given interval. Suppose $a$ and $b$ are two real numbers such that.

(a) $f(x)$ and its first derivative $f'(x)$ are continuous for $a \leq x \leq b$.

(b) $f(a)$ and $f(b)$ have opposite signs.

(c) $f'(x)$ is different from zero for all values of $x$ between $a$ and $b$.

Then there is one and only one root of the equation $f(x) = 0$ in $(a, b)$.

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