What is the primary difference between an increasing function and a strictly increasing function?

An increasing function satisfies $f'(x) \ge 0$, allowing for 'flat' sections where the gradient is zero. A strictly increasing function requires $f'(x) > 0$, meaning the function must always be moving upwards without any flat plateaus.

How does the condition $f(x) > 0$ differ from the condition $f'(x) > 0$?

$f(x) > 0$ means the function's output is positive (the graph is above the x-axis). $f'(x) > 0$ means the function's gradient is positive (the graph is sloping upwards), regardless of whether the actual y-values are positive or negative.

When finding intervals of decrease, why is it helpful to sketch the graph of $y = f'(x)$?

Sketching $f'(x)$ allows you to visually identify the regions where the derivative is below the x-axis ($f'(x) < 0$). This provides a clear geometric representation of the intervals where the original function $f(x)$ is decreasing.

What is a common mistake when solving quadratic inequalities like $x^2 - 9 < 0$ to find decreasing intervals?

A common error is failing to recognize that the solution is a single interval between the roots (e.g., $-3 < x < 3$). Students often incorrectly split it into two separate inequalities like $x < 3$ and $x < -3$.

Why might a student incorrectly conclude a function is increasing everywhere if they only check $f(x)$ values?

If a student only checks that $f(x)$ is positive, they are ignoring the rate of change. A function can have very high positive values but still be decreasing if those values are getting smaller as $x$ increases.

What happens to the inequality sign if you divide by a negative coefficient while solving for an increasing interval?

The inequality sign must be reversed. For example, if $-2x > 10$, then $x < -5$. Forgetting this flip will result in identifying the wrong interval for the function's behavior.

Define a 'decreasing function' in terms of its derivative.

A function $f(x)$ is decreasing on an interval if its first derivative $f'(x)$ is less than or equal to zero ($f'(x) \le 0$) for all values of $x$ in that interval.

What does the term 'monotonic' mean in the context of functions?

A monotonic function is one that is either entirely non-increasing or entirely non-decreasing over its entire domain. It does not change direction.

What is the derivative condition for a function to be stationary at a point?

A function is stationary at a point if its first derivative is equal to zero ($f'(x) = 0$). This indicates a horizontal tangent where the function is neither increasing nor decreasing.

If $f'(x) = 3x^2 + 5$, why is the function $f(x)$ strictly increasing for all real values of $x$?

Since $x^2$ is always $\ge 0$, $3x^2 + 5$ will always be $\ge 5$. Because the derivative is always strictly greater than zero, the function never stops increasing.

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Increasing & Decreasing Functions

Summary

The study of increasing and decreasing functions involves using the first derivative to determine the intervals over which a function's value rises or falls. By analyzing the sign of the gradient, mathematicians can describe the monotonic behavior of a curve and identify critical transition points.

1. Definition & Core Concepts

A graph showing a curve with shaded regions indicating where the function is increasing (positive gradient) and decreasing (negative gradient).

2. Underlying Principles

3. Methods & Techniques

4. Key Distinctions

5. Exam Strategy & Tips

Sketch the Derivative: If the derivative is a quadratic, sketching its parabola helps visualize where the gradient is positive (above the x-axis) or negative (below the x-axis).
Check Endpoints: When asked for an interval, ensure you know whether the question requires strict inequalities ( $<, >$ ) or non-strict ones ( $\le, \ge$ ).
Verify with Coordinates: Pick a test value within your calculated interval and substitute it into $f'(x)$ to ensure the sign matches your conclusion.
Read the Question Carefully: Distinguish between 'Find the values of $x$ for which $f(x)$ is increasing' and 'Show that $f(x)$ is increasing for all $x$ '.

6. Common Pitfalls & Misconceptions

7. Connections & Extensions

Increasing & Decreasing Functions

Q: What is a common mistake when solving quadratic inequalities like $x^2 - 9 < 0$ to find decreasing intervals?

A common error is failing to recognize that the solution is a single interval between the roots (e.g., $-3 < x < 3$). Students often incorrectly split it into two separate inequalities like $x < 3$ and $x < -3$.

Q: Why might a student incorrectly conclude a function is increasing everywhere if they only check $f(x)$ values?

If a student only checks that $f(x)$ is positive, they are ignoring the rate of change. A function can have very high positive values but still be decreasing if those values are getting smaller as $x$ increases.

Q: What happens to the inequality sign if you divide by a negative coefficient while solving for an increasing interval?

The inequality sign must be reversed. For example, if $-2x > 10$, then $x < -5$. Forgetting this flip will result in identifying the wrong interval for the function's behavior.

Q: Define a 'decreasing function' in terms of its derivative.

A function $f(x)$ is decreasing on an interval if its first derivative $f'(x)$ is less than or equal to zero ($f'(x) \le 0$) for all values of $x$ in that interval.

Q: What does the term 'monotonic' mean in the context of functions?

A monotonic function is one that is either entirely non-increasing or entirely non-decreasing over its entire domain. It does not change direction.

Q: What is the derivative condition for a function to be stationary at a point?

A function is stationary at a point if its first derivative is equal to zero ($f'(x) = 0$). This indicates a horizontal tangent where the function is neither increasing nor decreasing.

Q: If $f'(x) = 3x^2 + 5$, why is the function $f(x)$ strictly increasing for all real values of $x$?

Since $x^2$ is always $\ge 0$, $3x^2 + 5$ will always be $\ge 5$. Because the derivative is always strictly greater than zero, the function never stops increasing.

Summary

1. Definition & Core Concepts

A function $f(x)$ is defined as increasing on an interval if, for any two numbers $x_1$ and $x_2$ in that interval, $x_1 < x_2$ implies $f(x_1) \le f(x_2)$ . Graphically, this means the curve moves upwards or stays flat as it moves from left to right.

A function is strictly increasing if $x_1 < x_2$ implies $f(x_1) < f(x_2)$ , meaning the function never flattens out and always gains value.

Conversely, a function is decreasing if $x_1 < x_2$ implies $f(x_1) \ge f(x_2)$ , and strictly decreasing if $f(x_1) > f(x_2)$ . This corresponds to a downward slope on a Cartesian plane.

The monotonicity of a function refers to whether it is entirely increasing or entirely decreasing over a specific domain.

A graph showing a curve with shaded regions indicating where the function is increasing (positive gradient) and decreasing (negative gradient).

2. Underlying Principles

The First Derivative $f'(x)$ represents the instantaneous rate of change or the gradient of the tangent to the curve at any point $x$ .

If $f'(x) > 0$ for all $x$ in an interval, the function is strictly increasing because the positive gradient indicates the output $y$ is growing relative to $x$ .

If $f'(x) < 0$ for all $x$ in an interval, the function is strictly decreasing because the negative gradient indicates the output $y$ is shrinking as $x$ increases.

At points where $f'(x) = 0$ , the function is stationary. These points often act as the boundaries between increasing and decreasing intervals.

3. Methods & Techniques

Step 1: Differentiate: Find the first derivative $f'(x)$ of the given function using standard differentiation rules.
Step 2: Set up Inequalities: To find where the function is increasing, set $f'(x) \ge 0$ . To find where it is decreasing, set $f'(x) \le 0$ .
Step 3: Solve the Inequality: Determine the range of $x$ values that satisfy the inequality. This often involves solving quadratic or polynomial inequalities.
Step 4: Define Intervals: Express the final answer using interval notation (e.g., $x \in [a, b]$ ) or inequality notation (e.g., $a \le x \le b$ ).

4. Key Distinctions

It is vital to distinguish between the value of the function $f(x)$ and the value of its derivative $f'(x)$ . A function can be positive ( $f(x) > 0$ ) while it is decreasing ( $f'(x) < 0$ ).

Term	Condition	Visual Description
Increasing	$f'(x) \ge 0$	Moving uphill or flat
Strictly Increasing	$f'(x) > 0$	Moving uphill only
Decreasing	$f'(x) \le 0$	Moving downhill or flat
Strictly Decreasing	$f'(x) < 0$	Moving downhill only

5. Exam Strategy & Tips

Sketch the Derivative: If the derivative is a quadratic, sketching its parabola helps visualize where the gradient is positive (above the x-axis) or negative (below the x-axis).
Check Endpoints: When asked for an interval, ensure you know whether the question requires strict inequalities ( $<, >$ ) or non-strict ones ( $\le, \ge$ ).
Verify with Coordinates: Pick a test value within your calculated interval and substitute it into $f'(x)$ to ensure the sign matches your conclusion.
Read the Question Carefully: Distinguish between 'Find the values of $x$ for which $f(x)$ is increasing' and 'Show that $f(x)$ is increasing for all $x$ '.

6. Common Pitfalls & Misconceptions

Confusing $f(x)$ with $f'(x)$ : Students often solve $f(x) > 0$ (where the graph is above the x-axis) instead of $f'(x) > 0$ (where the graph is sloping upwards).
Algebraic Errors in Inequalities: When solving $f'(x) < 0$ , remember that multiplying or dividing an inequality by a negative number flips the inequality sign.
Ignoring the Domain: Some functions have restricted domains (like $\ln(x)$ or $\sqrt{x}$ ); intervals of increase/decrease must exist within these valid domains.

7. Connections & Extensions

This concept is the foundation for finding Stationary Points. A point where a function changes from increasing to decreasing is a local maximum, while a change from decreasing to increasing is a local minimum.

In real-world Optimization problems, identifying increasing intervals helps determine when a variable (like profit or velocity) is improving over time.

A function is strictly increasing if $x_1 < x_2$ implies $f(x_1) < f(x_2)$ , meaning the function never flattens out and always gains value.

Conversely, a function is decreasing if $x_1 < x_2$ implies $f(x_1) \ge f(x_2)$ , and strictly decreasing if $f(x_1) > f(x_2)$ . This corresponds to a downward slope on a Cartesian plane.

The monotonicity of a function refers to whether it is entirely increasing or entirely decreasing over a specific domain.

The First Derivative $f'(x)$ represents the instantaneous rate of change or the gradient of the tangent to the curve at any point $x$ .

If $f'(x) > 0$ for all $x$ in an interval, the function is strictly increasing because the positive gradient indicates the output $y$ is growing relative to $x$ .

If $f'(x) < 0$ for all $x$ in an interval, the function is strictly decreasing because the negative gradient indicates the output $y$ is shrinking as $x$ increases.

At points where $f'(x) = 0$ , the function is stationary. These points often act as the boundaries between increasing and decreasing intervals.

Step 1: Differentiate: Find the first derivative $f'(x)$ of the given function using standard differentiation rules.
Step 2: Set up Inequalities: To find where the function is increasing, set $f'(x) \ge 0$ . To find where it is decreasing, set $f'(x) \le 0$ .
Step 3: Solve the Inequality: Determine the range of $x$ values that satisfy the inequality. This often involves solving quadratic or polynomial inequalities.
Step 4: Define Intervals: Express the final answer using interval notation (e.g., $x \in [a, b]$ ) or inequality notation (e.g., $a \le x \le b$ ).

It is vital to distinguish between the value of the function $f(x)$ and the value of its derivative $f'(x)$ . A function can be positive ( $f(x) > 0$ ) while it is decreasing ( $f'(x) < 0$ ).

Term	Condition	Visual Description
Increasing	$f'(x) \ge 0$	Moving uphill or flat
Strictly Increasing	$f'(x) > 0$	Moving uphill only
Decreasing	$f'(x) \le 0$	Moving downhill or flat
Strictly Decreasing	$f'(x) < 0$	Moving downhill only

Confusing $f(x)$ with $f'(x)$ : Students often solve $f(x) > 0$ (where the graph is above the x-axis) instead of $f'(x) > 0$ (where the graph is sloping upwards).
Algebraic Errors in Inequalities: When solving $f'(x) < 0$ , remember that multiplying or dividing an inequality by a negative number flips the inequality sign.
Ignoring the Domain: Some functions have restricted domains (like $\ln(x)$ or $\sqrt{x}$ ); intervals of increase/decrease must exist within these valid domains.