Unit 5: Calculus

Topic 5.7: The Second Derivative Questions

Practice 20 exam-style questions for IB Math AA SL Topic 5.7. Review the question stems below, then unlock the full Question Bank to access markschemes, model answers, and AI grading.

1Find2 marks

2026

Given

f (x) = 2 x^{3} - 5 x + 1

, find f''(x).

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2Find3 marks

2024

For the family of curves

y = x^{4} - 2 a x^{3} + a^{2} x

, where

a

is a non-zero constant, find

\frac{d ^{2} y}{d x ^{2}}

in terms of

a

and

x

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3Choose1 mark

A curve has a stationary point at

x = 4

, where

f^{''} (4) = - 6

. State what kind of point it is.

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4Choose1 mark

Given

f (x) = 4 x^{3} - 7 x^{2} + 2 x

, find

f^{''} (x)

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5State2 marks

2026

The function

g (x) = 2 x^{3} + 3 x^{2} - 36 x

has stationary points at

x = - 3

and

x = 2

. State, with a reason, the nature of the stationary point at

x = - 3

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6Find2 marks

2026

Given f(x) = 2x³ − 5x + 1, find f''(x).

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7State2 marks

2026

The curve

y = x^{3} - 3 x^{2} - 9 x + 1

has a horizontal tangent where

x = 3

. By considering the second derivative, state the nature of this stationary point, giving a reason.

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8State2 marks

2026

A curve has a stationary point at x = 2, where f''(2) = −5. State the nature of this point.

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9State2 marks

2026

A curve has a stationary point at

x = 2

, where

f^{''} (2) = - 5

. State the nature of this point.

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10Find3 marks

2025

Let

f (x) = e^{x} - x^{3}

. Find the intervals on which the curve

y = f (x)

is concave up, giving the end points to three significant figures.

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11Show that3 marks

2021

A system is given by

\frac{d x}{d t} = y

and

\frac{d y}{d t} = - 2 x + 3 y

. Show that

\frac{d ^{2} y}{d t ^{2}} - 3 \frac{d y}{d t} + 2 y = 0

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12Show that4 marks

2026

Show that

f^{''} (x) = 12 x^{2} - 12

, and hence that the curve is concave up for

∣ x ∣ > 1

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13Show that3 marks

2022

Let

y = x e^{- x}

. Show that

\frac{d ^{2} y}{d x ^{2}} + 2 \frac{d y}{d x} + y = 0

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14Choose1 mark

A stationary point of a curve is at

x = - 2

, where

f^{''} (- 2) = 0

. Explain why the second-derivative test cannot classify it.

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15Choose1 mark

For

f (x) = x^{3} - 9 x^{2}

, find the values of

x

where the curve is concave up.

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16Choose1 mark

The curve

y = 2 x^{3} - 15 x^{2} + 4 x

has one point of inflexion. Find its

x

-coordinate.

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17Show that2 marks

2026

Show that

f^{''} (x) = 6 x - 12

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18Find4 marks

2026

Find the

x

-coordinates of the stationary points and use the second derivative to classify each.

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19Show that3 marks

2021

Two quantities satisfy

\frac{d x}{d t} = 3 x + y

and

\frac{d y}{d t} = 2 x

. Show that

\frac{d ^{2} y}{d t ^{2}} - 3 \frac{d y}{d t} - 2 y = 0

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20Find2 marks

2026

A curve has

f^{'} (x) = 3 x^{2} - 6 x

. Find f''(x), and use it to classify the stationary point at

x = 2

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