IB Mathematics AA Increasing and decreasing functions Study Notes
IB Mathematics AA Increasing and decreasing functions Study Notes
IB Mathematics AA Increasing and decreasing functions Notes Offer a clear explanation of Use of Increasing and decreasing functions, including various formula, rules, exam style questions as example to explain the topics. Worked Out examples and common problem types provided here will be sufficient to cover for topic Increasing and decreasing functions.
Increasing and Decreasing Functions
Increasing and Decreasing Functions
The derivative of a function tells us whether the function is increasing or decreasing at a point or over an interval.
Increasing function:
A function \( f(x) \) is increasing on an interval if \( f'(x) > 0 \) for all \( x \) in that interval. The graph rises as \( x \) increases.
Decreasing function:
A function \( f(x) \) is decreasing on an interval if \( f'(x) < 0 \) for all \( x \) in that interval. The graph falls as \( x \) increases.
If \( f'(x) = 0 \) at isolated points, these could be stationary points (maximum, minimum or point of inflection).
Example:
Determine where \( f(x) = x^3 – 3x^2 \) is increasing or decreasing.
▶️ Answer/Explanation
\( f'(x) = 3x^2 – 6x = 3x(x – 2) \)
critical points
\( f'(x) = 0 \Rightarrow x = 0 \) or \( x = 2 \)
Test intervals
- \( x < 0 \): \( f'(-1) = 3(-1)(-3) = 9 > 0 \) → increasing
- \( 0 < x < 2 \): \( f'(1) = 3(1)(-1) = -3 < 0 \) → decreasing
- \( x > 2 \): \( f'(3) = 3(3)(1) = 9 > 0 \) → increasing
Conclusion:
- Increasing on \( (-\infty, 0) \) and \( (2, \infty) \)
- Decreasing on \( (0, 2) \)
Example:
Identify the intervals where \( f(x) = x^4 – 4x^3 + 6x^2 \) is increasing or decreasing.
▶️ Answer/Explanation
\( f'(x) = 4x^3 – 12x^2 + 12x \)
\( f'(x) = 4x(x^2 – 3x + 3) \)
Solve \( f'(x) = 0 \)
\( 4x = 0 \Rightarrow x = 0 \)
\( x^2 – 3x + 3 = 0 \)
Discriminant: \( (-3)^2 – 4(1)(3) = 9 – 12 = -3 \) (no real roots)
So, critical point only at \( x = 0 \).
Test intervals
- \( x < 0 \): e.g., \( f'(-1) = 4(-1)((-1)^2 – 3(-1) + 3) = -4(1 + 3 + 3) = -4(7) = -28 < 0 \) → decreasing
- \( x > 0 \): \( f'(1) = 4(1)((1)^2 – 3(1) + 3) = 4(1)(1 – 3 + 3) = 4(1)(1) = 4 > 0 \) → increasing
Conclusion:
- Decreasing on \( (-\infty, 0) \)
- Increasing on \( (0, \infty) \)
The function decreases up to \( x = 0 \) and increases after that. The point at \( x=0 \) is a local minimum.
Graphical Interpretation of Derivative Sign
Graphical Interpretation of Derivative Sign
The sign of the derivative \( f'(x) \) tells us about the slope of the tangent and how the function behaves at each point:
- \( f'(x) > 0 \): The function is increasing. The graph rises as you move from left to right. The tangent lines have positive slopes.
- \( f'(x) = 0 \): The function has a horizontal tangent at that point. This could indicate a stationary point (maximum, minimum, or point of inflection).
- \( f'(x) < 0 \): The function is decreasing. The graph falls as you move from left to right. The tangent lines have negative slopes.
These conditions help identify where a function is increasing, decreasing, or has stationary points.
Example:
Consider \( f(x) = -x^2 + 4x \).
Find where the function is increasing, decreasing, and identify stationary points.
▶️ Answer/Explanation
\( f'(x) = -2x + 4 \)
Solve \( f'(x) = 0 \)
\( -2x + 4 = 0 \Rightarrow x = 2 \)
Test intervals
- \( x < 2 \): \( f'(1) = -2(1) + 4 = 2 > 0 \) → increasing
- \( x > 2 \): \( f'(3) = -2(3) + 4 = -2 < 0 \) → decreasing
Conclusion:
- Function increases on \( (-\infty, 2) \)
- Function decreases on \( (2, \infty) \)
- Stationary point at \( x = 2 \) → maximum
