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Study AHL 5.10 (HL)—Second derivative and concavity conceptsOpen the corresponding Concept lesson for explanations, examples and exam connections.

[Maximum number: 3]

Kailash manufactures drink containers in the shape of a cuboid. The container has a square top and a square base of length, $l\,\mathrm{cm}$ . Its height, $d\,\mathrm{cm}$ , is three times the length of the base.

diagram not to scale

(a)

Find $\frac{\mathrm{d}^{2} A}{\mathrm{~d} r^{2}}$ .

[ 1 ]

$\left(\frac{\mathrm{d}^{2} A}{\mathrm{~d} r^{2}}=\right) 4 \pi+\frac{1500}{r^{3}}$

(b)

Hence determine whether the graph of A is concave-up or concave-down for r>0. Justify your answer.

[ 2 ]

As r>0, $\frac{\mathrm d^2A}{\mathrm d r^2}>0$ .
Hence the graph is concave-up.
Note: marks in part (ii) require an expression seen in part (i).

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IB Maths AI HL Ahl 5 10 Hl Second Derivative and Concavity Topic Practice

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Question 1

Question (a)

Question (b)