IB Mathematics AI SL Increasing and decreasing function MAI Study Notes - New Syllabus

IB Mathematics AI SL Increasing and decreasing function MAI Study Notes

LEARNING OBJECTIVE

Key Concepts:

Increasing and Decreasing Functions

The derivative \( f'(x) \) gives the rate of change (slope) of a function at any point.

If \( f'(x) > 0 \), the function is increasing on that interval.
If \( f'(x) < 0 \), the function is decreasing on that interval.
If \( f'(x) = 0 \), the function may have a local maximum, local minimum, or a stationary point.
This information helps identify where a function is rising, falling, or flat, based on its first derivative.

Example

Let \( f(x) = x^3 – 3x^2 + 2 \). Determine where the function is increasing and decreasing.

▶️ Answer/Explanation

Solution:

Find the first derivative:
\( f'(x) = 3x^2 – 6x \)
Factor the derivative:
\( f'(x) = 3x(x – 2) \)
Solve \( f'(x) = 0 \):
Critical points at \( x = 0 \) and \( x = 2 \)
Test intervals around the critical points:
- For \( x < 0 \), pick \( x = -1 \): \( f'(-1) = 3(-1)(-3) = 9 > 0 \) → Increasing
- For \( 0 < x < 2 \), pick \( x = 1 \): \( f'(1) = 3(1)(-1) = -3 < 0 \) → Decreasing
- For \( x > 2 \), pick \( x = 3 \): \( f'(3) = 3(3)(1) = 9 > 0 \) → Increasing
📈 The function is increasing on \( (-\infty, 0) \cup (2, \infty) \), and decreasing on \( (0, 2) \).

Graphical Interpretation of \( f'(x) > 0 \), \( f'(x) = 0 \), \( f'(x) < 0 \)

The derivative \( f'(x) \) describes the slope of the tangent to the graph of a function at any point \( x \).

\( f'(x) > 0 \): The function is increasing at that point — graph slopes upward.
\( f'(x) < 0 \): The function is decreasing at that point — graph slopes downward.
\( f'(x) = 0 \): The function has a horizontal tangent — could be a maximum, minimum, or point of inflection.
This information can be used to understand the behavior of a function just by analyzing its derivative.

Example

Using the graph of a function \( f(x) =x^3-3x^2+2\), describe the behavior of \( f'(x) \) in terms of positive, zero, and negative values.

▶️ Answer/Explanation

Solution:

Between \( x = -3 \) and \( x = -1 \), the graph is rising → \( f'(x) > 0 \)
At \( x = -1 \), the graph has a local maximum → \( f'(x) = 0 \)
Between \( x = -1 \) and \( x = 2 \), the graph is falling → \( f'(x) < 0 \)
At \( x = 2 \), the graph has a local minimum → \( f'(x) = 0 \)
For \( x > 2 \), the graph is rising again → \( f'(x) > 0 \)
This shows how the sign of the derivative corresponds to the slope of the function at different intervals.

Identifying Intervals of Increase and Decrease

To determine where a function is increasing or decreasing, we use its first derivative \( f'(x) \).

Steps:
1. Find the first derivative \( f'(x) \).
2. Solve \( f'(x) = 0 \) to get the critical points.
3. Test intervals between the critical points by plugging values into \( f'(x) \).
4. If \( f'(x) > 0 \), the function is increasing on that interval.
5. If \( f'(x) < 0 \), the function is decreasing on that interval.
This helps understand the shape of the function and locate relative maxima and minima.

Example

Determine the intervals where the function \( f(x) = x^3 – 6x^2 + 9x \) is increasing or decreasing.

▶️ Answer/Explanation

Find the first derivative:
\( f'(x) = 3x^2 – 12x + 9 \)
Solve \( f'(x) = 0 \):
\( 3x^2 – 12x + 9 = 0 \Rightarrow x = 1 \text{ and } x = 3 \)
Test intervals:
- For \( x < 1 \), pick \( x = 0 \):
  \( f'(0) = 9 > 0 \) → Increasing
- For \( 1 < x < 3 \), pick \( x = 2 \):
  \( f'(2) = -3 < 0 \) → Decreasing
- For \( x > 3 \), pick \( x = 4 \):
  \( f'(4) = 9 > 0 \) → Increasing
Increasing on: \( (-\infty, 1) \cup (3, \infty) \)
Decreasing on: \( (1, 3) \)