International Baccalaureate Mathematics

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:

"Consider the cubic function $f(x)=\frac{1}{6}x^3-2x^2+6x-2$" ... more

Click on a topic below for suggested lesson Starters, resources and activities from Transum.

Calculus It is said that the word calculus comes from the Latin word for the small pebble used for counting and calculations. The two major branches, differentiation and integration, are studied by pupils only towards the end of their school days but does then form a major part of their studies. A course in calculus is a prerequisite for other, more advanced courses in mathematical analysis. Isaac Newton is best known for developing calculus. He was born in England in 1642 and is considered one of the most influential scientists and mathematicians in history.

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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