IB Maths AA HL 5.7 Second derivative and graph behaviour Question Bank

Practise IB Mathematics SL/HL 5.7 by applying second derivative and graph behaviour methods to exam-style questions.

Exam points

identify the mathematical structure, variable or representation
select and apply the correct theorem, formula or algorithm
check the result using units, domain, graph or logical reasoning

[Maximum number: 3]

This question investigates a ratio of lengths found from the line passing through the points of inflexion of a quartic curve of the form $y=x^{4}-m x^{3}+n x$ .
The curve $y=x^{4}-3 x^{3}+3 x$ has points of inflexion at B and C . The line passing through B and C intersects the curve again at points A and D . This is shown in the following graph.

Figure for Question SL 5.7—Second derivative and graph behaviour question 1 — IB Maths AA HL

Find $\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}$ .

$\frac{\mathrm{d} y}{\mathrm{~d} x}=4 x^{3}-9 x^{2}+3$
(M1)(A1)

Note: Award M1 for at least two correct terms.

\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=12 x^{2}-18 x

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SL 5.7—Second derivative and graph behaviour question 1