Second Derivative and Concavity (AA SL)
Differentiate a function twice and you learn something new about its shape: whether the curve bends upward like a cup or downward like a dome, and where it switches between the two. This page covers finding \(f''(x)\) and using it to locate points of inflexion. It's part of the broader Differentiation topic.
11 questions on this sub-topic.
Second derivative and concavity
Covered under IB syllabus references SL5.7 (the second derivative, and the graphical relationship between \(f\), \(f'\) and \(f''\)) and SL5.8 (points of inflexion, and the concave-up/concave-down terminology).
Second derivative
\(f''(x)\) is found by differentiating \(f'(x)\) using the same rules a second time.
Written as \(\dfrac{d^2y}{dx^2}\) in Leibniz notation.
Point of inflexion
Solve \(f''(x)=0\), then confirm concavity changes sign either side of that \(x\)-value.
\(f''(x)>0\) means concave up; \(f''(x)<0\) means concave down.
Need the full differentiation syllabus and GDC tools for checking concavity graphically? See Differentiation.
Worked examples
Given \(y = x^{3} + 2x\), find \(\dfrac{d^{2}y}{dx^{2}}\).
Worked solution
\(\dfrac{dy}{dx}=3x^2+2.\) A1
\(\dfrac{d^2y}{dx^2}=6x.\) M1 A1
For \(f(x) = x^3 - 3x^2\):
(a) Find \(f''(x)\).
(b) Find the \(x\)-coordinate of the point of inflexion.
Worked solution
(a) Differentiate. \(f'(x)=3x^2-6x\) A1
so \(f''(x)=6x-6.\) A1
(b) A point of inflexion occurs where \(f''=0\) and concavity changes sign. \(6x-6=0\Rightarrow x=1;\) M1
since \(f''\) changes from negative to positive there, it is genuine. A1
For \(f(x) = x^4 - 4x^3 + 6x^2\):
(a) Find \(f''(x)\).
(b) Find all points of inflexion, justifying each.
Worked solution
(a) \(f'(x) = 4x^3 - 12x^2 + 12x;\) \(f''(x) = 12x^2 - 24x + 12\) M1
\(= 12(x-1)^2.\) A1
(b) \(f''(x) = 0 \Rightarrow x = 1.\) M1
But \(f''(x) = 12(x-1)^2 \ge 0\) - no sign change at \(x = 1\). Therefore there is no point of inflexion. A1R1
For \(f(x)=x^3-6x^2+5\):
(a) Find \(f''(x).\)
(b) Find the point of inflexion.
Worked solution
(a) \(f'(x) = 3x^2 - 12x,\) M1
\(f''(x) = 6x - 12.\) A1
(b) \(f''(x) = 0 \Rightarrow x = 2.\) M1
\(f(2) = -11,\) point \((2, -11).\) A1 R1
Common mistakes
- Stopping after one differentiation. Finding \(f''(x)\) means differentiating twice - a quick way to lose easy marks is to hand in \(f'(x)\) and call it done.
- Assuming \(f''(x)=0\) always gives a point of inflexion. It only marks a candidate - the concavity has to actually change sign either side of that \(x\)-value, otherwise it's not a genuine inflexion point.
- Mixing up the signs of concavity. \(f''(x)>0\) means the curve is concave up (like a cup); \(f''(x)<0\) means concave down (like a dome) - it's easy to remember these the wrong way round.
Ready to practise properly?
10 second-derivative-and-concavity questions, marked instantly like the real exam.
Quick answers
How do I find the second derivative of a function?
Differentiate the function once to get \(f'(x)\), then differentiate that result again using the same rules to get \(f''(x)\).
How do I find a point of inflexion?
Solve \(f''(x) = 0\), then confirm the concavity actually changes sign either side of that \(x\)-value - if it does, it is a genuine point of inflexion.