Understanding Increasing and Decreasing Functions

A function is described as increasing or decreasing depending on the manner in which its output values change as the input variable progresses. The classification as "monotonic" arises when the function preserves order over intervals, a fundamental idea in calculus and analysis.

Formal Definition of Increasing Functions on an Interval

Let $I \subseteq \mathbb{R}$ be an interval, and let $f: I \to \mathbb{R}$ be a real-valued function. The function $f$ is said to be increasing on $I$ if for any $x_1, x_2 \in I$ with $x_1 < x_2$, it holds that $f(x_1) \leq f(x_2)$.

If the stronger inequality $f(x_1) < f(x_2)$ holds for all $x_1, x_2 \in I$ with $x_1 < x_2$, then $f$ is termed strictly increasing on $I$.

Formal Definition of Decreasing Functions on an Interval

A function $f: I \to \mathbb{R}$ is called decreasing on the interval $I$ if for any $x_1, x_2 \in I$ such that $x_1 < x_2$, one has $f(x_1) \geq f(x_2)$.

If $f(x_1) > f(x_2)$ for every $x_1, x_2 \in I$ with $x_1 < x_2$, then $f$ is strictly decreasing on $I$.

First Derivative Test for Monotonicity in Differentiable Functions

Let $f: I \to \mathbb{R}$ be differentiable on the open interval $I$. The sign of the derivative $f'(x)$ determines the monotonic character of $f$:

(i) Increasing: If $f'(x) \geq 0$ for all $x \in I$, then $f$ is increasing on $I$. If, moreover, $f'(x) > 0$ for all $x \in I$, then $f$ is strictly increasing.

(ii) Decreasing: If $f'(x) \leq 0$ for all $x \in I$, then $f$ is decreasing on $I$. If $f'(x) < 0$ for every $x \in I$, then $f$ is strictly decreasing.

This conclusion draws from the mean value theorem, ensuring that the derivative’s sign controls the direction of change of $f(x)$ over every subinterval of $I$. For the underlying proof and further mean value theorem applications, refer to Differential Calculus.

Derivation: Method to Determine Intervals of Increase and Decrease

To determine on which intervals a differentiable function $f(x)$ is increasing or decreasing:

1. Compute the derivative $f'(x)$.

2. Solve $f'(x) = 0$ to find critical points.

3. Identify the sign of $f'(x)$ in the intervals delimited by these critical points.

4. Each interval where $f'(x) > 0$ is an interval of strict increase, and where $f'(x) < 0$ is an interval of strict decrease.

Explicit Example: Monotonicity of $f(x) = x^3$ for $x \in \mathbb{R}$

Given: $f(x) = x^3$ for all $x \in \mathbb{R}$.

Step 1 (Derivative): $f'(x) = \dfrac{d}{dx} (x^3)$.

$f'(x) = 3x^2$

Step 2 (Sign of $f'(x)$): Since $x^2 \geq 0$ for all $x \in \mathbb{R}$, it follows that $f'(x) \geq 0$ for all $x \in \mathbb{R}$.

Final result: Thus, $f(x) = x^3$ is increasing everywhere on $\mathbb{R}$. Since $f'(x) = 0$ only at $x = 0$ (and positive elsewhere), it is strictly increasing except at $x = 0$.

Explicit Example: Monotonicity of $f(x) = xe^{-x}$

Given: $f(x) = x e^{-x}$, $x \in \mathbb{R}$.

Step 1 (Derivative): Use the product rule:

$\dfrac{d}{dx}(x e^{-x}) = \dfrac{d}{dx}(x)\cdot e^{-x} + x \cdot \dfrac{d}{dx}(e^{-x})$

$= 1 \cdot e^{-x} + x \cdot (-e^{-x})$

$= e^{-x} - x e^{-x}$

$= e^{-x}(1 - x)$

Step 2 (Critical point): Set $f'(x) = 0$.

$e^{-x}(1 - x) = 0$

Since $e^{-x}$ is never zero, set $1 - x = 0$.

$x = 1$

Step 3 (Sign analysis): For $x < 1$, $1 - x > 0$ and $e^{-x} > 0$, so $f'(x) > 0$. For $x > 1$, $1 - x < 0$, so $f'(x) < 0$.

Final result: $f(x)$ is increasing on $(-\infty, 1)$ and decreasing on $(1, \infty)$.

Review more advanced examples and variations at Derivative Examples.

Relation Between Monotonicity and Constant Functions

A function $f(x)$ is said to be constant on $I$ if $f(x_1) = f(x_2)$ for all $x_1, x_2 \in I$. In this case, $f'(x) = 0$ for all $x \in I$, and $f$ is both increasing and decreasing (in the weak sense), but not strictly monotonic.

Algebraic Properties of Increasing and Decreasing Functions

If $f$ and $g$ are both increasing on $I$, their sum $f + g$ is increasing on $I$. If $f$ and $g$ are both decreasing, their sum $f + g$ is decreasing.

If $f$ is increasing on $I$, then $-f$ is decreasing, and vice versa.

If $f$ is increasing and positive on $I$, then the reciprocal function $\frac{1}{f(x)}$ is decreasing on $I$. If $f$ is decreasing and positive, then $\frac{1}{f(x)}$ is increasing.

Result: The product $f(x)g(x)$ of two increasing functions is increasing on $I$ if both $f, g \geq 0$ on $I$. For a full treatment of monotonicity and extrema, refer to Monotonicity And Extremum of Functions.

Graphical Interpretation of Increasing and Decreasing Functions

Graphically, if the plot of $y = f(x)$ rises (or remains flat) as $x$ increases, the function is increasing; if the graph falls (or remains flat), the function is decreasing. At points of local flatness (where $f'(x) = 0$), the function is constant over those intervals. Key graphical points correspond to solutions of $f'(x) = 0$ and intervals are then determined by derivative sign analysis.

Detailed Example: Monotonicity of $f(x) = \sin x$ on Intervals in $[0, \pi]$

Given: $f(x) = \sin x$, $x \in [0, \pi]$.

Step 1 (Derivative): $f'(x) = \cos x$

Step 2 (Sign of $f'(x)$): In $(0, \frac{\pi}{2})$, $\cos x > 0$; in $(\frac{\pi}{2}, \pi)$, $\cos x < 0$ and at $x = \frac{\pi}{2}$, $\cos x = 0$.

Step 3 (Conclusion): $f(x)$ is strictly increasing in $(0, \frac{\pi}{2})$, strictly decreasing in $(\frac{\pi}{2}, \pi)$, and stationary at $x = \frac{\pi}{2}$.

Problems involving monotonicity and their solutions can be found by studying Increasing And Decreasing Functions.

Typical Steps in Interval Testing for Monotonicity

For a differentiable function $f(x)$ on $(a, b)$:

1. Differentiate $f(x)$ to obtain $f'(x)$.

2. Solve $f'(x) = 0$ to obtain critical points.

3. Using these points, split the interval $(a, b)$ into subintervals.

4. For each subinterval, pick a test value $x_0$; determine the sign of $f'(x_0)$. If $f'(x_0) > 0$, $f(x)$ is strictly increasing on that subinterval; if $f'(x_0) < 0$, it is strictly decreasing.

Relation of Monotonicity to Optimization

Intervals where $f(x)$ is increasing or decreasing are essential in finding local or global maxima and minima. These concepts are deeply connected with the analysis of critical points, as discussed in the context of Maximum And Minimum Value of Quadratic Polynomial.

Conclusion on Increasing and Decreasing Functions

The behaviour of a differentiable function on an interval is entirely characterized by the sign of its first derivative. Increasing and decreasing functions, particularly their intervals, play a central role not only in theory but also in applications relevant to calculus, optimization, and mathematical modelling within and beyond the context of the JEE Main curriculum.