Increasing/Decreasing Functions explained through a friendly whiteboard conversation, examples, visuals, and practice for school students.
Step 1
Notes
Step 2
Formulae
Step 3
Examples
Step 4
Practice
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A function is increasing when its values go up as x increases.
It is decreasing when its values go down as x increases.
Derivatives help identify these intervals accurately.
The derivative gives the slope of the graph.
If f'(x) is positive on an interval, the function is increasing there.
If f'(x) is negative on an interval, the function is decreasing there.
Critical points occur where f' or where the derivative is not defined.
A sign table helps organise intervals around critical points.
Meaning: The graph rises as x moves right.
Definition: A function is increasing on an interval when larger alues give larger function values.
A positive derivative usually shows increasing behaviour.
Meaning: The graph falls as x moves right.
Definition: A function is decreasing on an interval when larger alues give smaller function values.
A negative derivative usually shows decreasing behaviour.
Meaning: Places where behaviour may change.
Definition: Critical points occur where f' or f'(x) is not defined.
They split the number line into intervals to test.
Meaning: Use derivative signs to classify intervals.
Definition: Test the sign of f'(x) in each interval around critical points.
Positive means increasing and negative means decreasing.
Easy
Question: If f' on (1,4), what happens to f on (1,4)?
Medium
Question: For , find where it is increasing.
Exam-level
Question: For x, find critical points.
Using f(x) sign instead of f'(x) sign
Why it happens: Function value and slope are confused.
Correct approach: Increasinecreasing depends on derivative sign.
Not splitting at critical points
Why it happens: Intervals are not organised.
Correct approach: Use critical points to make test intervals.
Including endpoints incorrectly
Why it happens: Open and closed intervals are mixed.
Correct approach: Use interval notation required by your curriculum and question.
Do I use f(x) or f'(x)?
Use f'(x), the derivative.
What is a critical point?
It is a point where the derivative is zero or not defined.
Can a function change from increasing to decreasing?
Yes, often near a maximum or minimum point.
Practise the same concept from this article. Your student profile may guide entitlement and follop recommendations, but it does not replace this concept.