IB Maths AA HL Sl 5 7 Second Derivative and Graph Behaviour Questions

Practise IB Mathematics HL 5.7 by applying second derivative and graph behaviour methods to questions from the Question Bank.

Exam points

Use second-derivative notation d²y/dx² and f″(x), calculating or interpreting f″ for a differentiated function.
Relate the graphs of f, f′ and f″ to stationary points, concavity and changes in gradient, using technology where it supports the interpretation.

[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 IB Maths AA HL Sl 5 7 Second Derivative and Graph Behaviour Questions 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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IB Maths AA HL Sl 5 7 Second Derivative and Graph Behaviour Questions question 1