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International Baccalaureate Mathematics

Calculus

Syllabus Content

Increasing and decreasing functions. Graphical interpretation of f'(x)>0, f'(x)=0,f'(x)<0

Here is an exam-style questions on this statement:

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Furthermore

Official Guidance, clarification and syllabus links:

Identifying intervals on which functions are increasing \((f'(x)\gt0)\) or decreasing \((f'(x)\lt 0)\).

An increasing function is one where the function values rise as the input values increase, while a decreasing function is one where the function values fall as the input values increase. The first derivative, \( f'(x) \), provides insight into the behavior of the function. Specifically:

1. If \( f'(x) > 0 \) for all \( x \) in an interval, then \( f(x) \) is increasing on that interval. This is because a positive derivative indicates a positive slope or rate of change.

2. If \( f'(x) = 0 \) at a point, it suggests that the function might have a horizontal tangent at that point, which could be a maximum, minimum, or a point of inflection.

3. If \( f'(x) < 0 \) for all \( x \) in an interval, then \( f(x) \) is decreasing on that interval. A negative derivative indicates a negative slope or rate of change.

Example: Consider the function \( f(x) = x^3 - 3x^2 + 2 \). Its derivative is given by:

$$ f'(x) = 3x^2 - 6x $$

To find the intervals where the function is increasing or decreasing, we set \( f'(x) = 0 \) and solve for \( x \):

$$ 3x^2 - 6x = 0 \\ x(3x - 6) = 0 \\ x = 0 \text{ or } x = 2 $$

Using the above values, we can determine the sign of \( f'(x) \) in each interval and thus the behavior of \( f(x) \).

For \( x < 0 \), \( f'(x) > 0 \) (increasing).

For \( 0 < x < 2 \), \( f'(x) < 0 \) (decreasing).

And for \( x > 2 \), \( f'(x) > 0 \) (increasing).

This video on Optimization and Calculus Curves is from Revision Village and is aimed at students taking the IB Maths AA SL/HL course.

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