Increasing and Decreasing Functions | JEE Main Calculus Crash Course | Mathematicon

Increasing and Decreasing Functions: Mastering Monotonicity for JEE Main

Hello JEE aspirants! Understanding increasing and decreasing functions is a cornerstone of calculus and a frequent topic in JEE Main. Mastering this concept allows you to analyze function behavior, solve optimization problems, and tackle curve sketching questions with confidence. Let's dive in!

What are Increasing and Decreasing Functions?

Intuitively, a function is increasing if its value goes up as you move from left to right along the x-axis. Conversely, a function is decreasing if its value goes down as you move from left to right. This property is called monotonicity.

Consider the function $f(x) = x^2$ . For $x < 0$ , as $x$ increases (e.g., from -3 to -2), $f(x)$ decreases (from 9 to 4). Thus, $f(x) = x^2$ is decreasing on the interval $(-\infty, 0)$ . For $x > 0$ , as $x$ increases (e.g., from 1 to 2), $f(x)$ increases (from 1 to 4). Thus, $f(x) = x^2$ is increasing on the interval $(0, \infty)$ . At $x=0$ , the function changes direction.

The First Derivative Test: Your Monotonicity Compass

The first derivative, $f'(x)$ , gives us the slope of the tangent line to the function $f(x)$ at any point. This slope is the key to determining whether a function is increasing or decreasing.

If $f'(x) > 0$ , it means the tangent line has a positive slope, indicating that the function is increasing at that point. Think of climbing a hill – you're going upwards! Similarly, if $f'(x) < 0$ , the tangent line has a negative slope, and the function is decreasing. You're going downhill!

Key Formulas Explained

Strictly Increasing Function

A function $f$ is strictly increasing on an interval $I$ if $f'(x) > 0$ for all $x \in I$ .

Explanation: For every $x$ in the interval $I$ , the slope of the tangent is positive, guaranteeing that the function's value consistently increases as $x$ increases.

Strictly Decreasing Function

A function $f$ is strictly decreasing on an interval $I$ if $f'(x) < 0$ for all $x \in I$ .

Explanation: The negative slope of the tangent ensures the function's value consistently decreases as $x$ increases.

For example, consider $f(x) = x^3$ . Its derivative is $f'(x) = 3x^2$ . Since $3x^2 \geq 0$ for all real numbers and $3x^2$ is only 0 at the single point $x=0$ , we can say $f(x) = x^3$ is strictly increasing everywhere. This is because it is increasing everywhere except at $x=0$ where it is momentarily flat (slope is 0).

Critical Points: Where the Magic Happens

Critical points are those points where the derivative of a function is either zero or undefined. These points are crucial because they often mark the transition between increasing and decreasing intervals. Think of them as potential turning points.

Mathematically, a critical point $c$ satisfies either $f'(c) = 0$ or $f'(c)$ is undefined. At $f'(c) = 0$ , the tangent line is horizontal. These points can indicate local maxima, local minima, or saddle points (points of inflection). If $f'(c)$ is undefined, this indicates a vertical tangent or a discontinuity.

For example, for $f(x) = |x|$ , the derivative is undefined at $x = 0$ . This is the critical point, and it's where the function changes from decreasing to increasing.

Tips for Solving Problems

Find the derivative: The first step is always to find $f'(x)$ .
Find critical points: Solve for $f'(x) = 0$ and identify where $f'(x)$ is undefined.
Create a sign chart: Draw a number line and mark all critical points. Choose test values within each interval created by the critical points and evaluate $f'(x)$ at these test values. The sign of $f'(x)$ in each interval tells you whether the function is increasing or decreasing.
Determine intervals: Based on the sign chart, identify the intervals where $f'(x) > 0$ (increasing) and where $f'(x) < 0$ (decreasing).

Let's say we want to find the intervals where $f(x) = x^3 - 3x$ is increasing or decreasing.

Find the derivative: $f'(x) = 3x^2 - 3$
Find critical points: Set $f'(x) = 0$ : $3x^2 - 3 = 0 \implies x^2 = 1 \implies x = \pm 1$ .

Create a sign chart:

Interval	Test Value	$f'(x) = 3x^2 - 3$	Sign	Conclusion
$(-\infty, -1)$	$x = -2$	$3(-2)^2 - 3 = 9$	$+$	Increasing
$(-1, 1)$	$x = 0$	$3(0)^2 - 3 = -3$	$-$	Decreasing
$(1, \infty)$	$x = 2$	$3(2)^2 - 3 = 9$	$+$	Increasing

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

Common Mistakes to Avoid

Forgetting critical points where $f'(x)$ is undefined: Don't just look for where $f'(x) = 0$ . Remember to check for points where the derivative doesn't exist (e.g., at sharp corners or vertical tangents).
Confusing $f'(x) \geq 0$ with strictly increasing: If $f'(x) \geq 0$ on an interval, the function is increasing but not strictly increasing. It could have flat spots where $f'(x) = 0$ . For strictly increasing, $f'(x)$ must be strictly greater than 0. Similarly, if $f'(x) \leq 0$ on an interval, the function is decreasing but not strictly decreasing.
Incorrectly determining intervals: Double-check your sign chart and make sure you're using the correct test values.
Assuming that a single point is an interval: A function is said to be increasing or decreasing *on an interval*. It makes no sense to say that a function is increasing or decreasing at a single point.

JEE-Specific Tricks

Visualization: For many JEE problems, sketching a rough graph of the function can provide valuable insights. Use the information about increasing and decreasing intervals to help you sketch the graph.
Optimization Problems: Understanding increasing and decreasing functions is essential for solving optimization problems (finding maxima and minima). The first derivative test helps identify candidate points for optimization.
Related Rates: In related rates problems, identifying whether a quantity is increasing or decreasing can help you determine the sign of its rate of change.
Connecting Concepts: Always try to connect increasing/decreasing functions to other calculus concepts like concavity, points of inflection, and limits. This holistic approach strengthens your understanding.

By understanding the key concepts, formulas, and tips discussed in this lesson, you'll be well-equipped to tackle JEE Main problems involving increasing and decreasing functions. Keep practicing, and you'll master this essential calculus topic!