Increasing or Decreasing Functions (AI SL)
Once you can differentiate, the sign of \(f'(x)\) tells you everything about the shape of the graph at that point: positive means the curve is climbing, negative means it's falling. This page focuses on that reading skill - working out and interpreting the sign of the derivative - with worked examples and the mistakes that catch students out. It's part of the broader Differentiation topic.
16 questions on this sub-topic.
Reading the sign of \(f'(x)\)
Covered under IB syllabus reference SL5.2. There's no formula to memorise here - it's about interpreting a number you've already found by differentiating.
Increasing
\[f'(x)>0\]
The curve is rising - as \(x\) increases, so does \(y\).
Decreasing
\[f'(x)<0\]
The curve is falling - as \(x\) increases, \(y\) decreases.
Need the full syllabus wording and GDC workflow for differentiation? See Differentiation.
Worked examples
For \(f(x) = x^2 - 6x + 5\):
(a) Find \(f'(x).\)
(b) Determine whether \(f\) is increasing or decreasing at \(x = 1.\)
Worked solution
(a) \(f'(x) = 2x - 6.\) M1 A1
(b) \(f'(1) = 2 - 6 = -4 < 0\), M1
\(f\) is decreasing at \(x = 1.\) A1
Find the intervals where \(f(x)=x^3-3x^2\) is increasing.
(a)(i) State one interval where \(f\) is increasing.
(a)(ii) State the other interval where \(f\) is increasing.
Worked solution
\(f'(x)=3x^2-6x=3x(x-2).\) M1
\(f'(x)>0\) when \(x<0\) or \(x>2.\) M1
Step 3 - \(x<0\).
A1 A1
At a point on a curve, \(f'(3)=5.\) State what this tells you about the function at that point.
Worked solution
\(f'(3)>0\), so the function is increasing at \(x=3\). A1
It is increasing at a rate of 5 units of \(y\) per unit of \(x\). A1
For \(f(x)=x^2-10x\):
(a) Find \(f'(x).\)
(b) Is \(f\) increasing or decreasing at \(x=2?\).
Worked solution
(a) \(f'(x)=2x-10.\) M1A1
(b) \(f'(2)=-6<0\), M1
decreasing. A1
Common mistakes
- Reading the sign of \(f(x)\) instead of \(f'(x)\). "Increasing" and "decreasing" describe the gradient, not the height of the curve - a function can have a large positive \(y\)-value while still decreasing at that point.
- Stopping after finding the derivative. Differentiating correctly is only step one; you then need to substitute the given \(x\)-value (or factorise and check signs) to actually answer the increasing/decreasing question.
- Missing one of two intervals in a cubic-type question. When \(f'(x)\) is a quadratic, it usually changes sign twice, giving two separate increasing (or decreasing) intervals - factorise fully and check both sides of each root rather than stopping at the first one found.
Ready to practise properly?
16 increasing/decreasing questions, marked instantly like the real exam.
Quick answers
How do you tell if a function is increasing or decreasing at a point?
Find the derivative and evaluate it at that point. If \(f'(x)>0\) the function is increasing there; if \(f'(x)<0\) it is decreasing.
What does \(f'(x) = 0\) mean?
The gradient is momentarily flat - the function is neither increasing nor decreasing at that exact point. This is where turning points occur.