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

Practise IB Mathematics HL 5.10 by applying second derivative and concavity methods to questions from the Question Bank.

Exam points

Calculate and interpret the second derivative to determine concavity and points of inflexion.
Use the second-derivative test with first-derivative information to classify maxima and minima.
Connect concavity or inflexion to the shape and meaning of an applied model.

[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 Questions question 1

Question (a)

Question (b)