Unit 5: Calculus

Topic 5.2: Increasing and Decreasing Functions Questions

Practice 20 exam-style questions for IB Math AA SL Topic 5.2. Review the question stems below, then unlock the full Question Bank to access markschemes, model answers, and AI grading.

1Deduce2 marks

2024

The rate of change of

y

with respect to

x

obeys

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

. Deduce that

y

rises everywhere as

x

increases.

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2Determine2 marks

2026

A function has

f^{'} (x) = 12 - 3 x

. Determine whether f is increasing or decreasing at

x = 5

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3Sketch1 mark

2026

Sketch a possible graph of

y = f (x)

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4Determine2 marks

2026

A function has f'(x) = 12 − 3x. Determine whether f is increasing or decreasing at x = 5.

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5Explain2 marks

2025

For x ≠ 2/3, a function f has gradient function f′(x) = −23/(3x − 2)².

Use the sign of f′(x) to explain why f is never increasing.

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6Explain2 marks

2025

Consider the function f(x) = x³ + 3x − 1, x ∈ ℝ.

Explain briefly why f is increasing for all values of x.

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7Explain2 marks

2025

A function f satisfies f′(x) = 7/(x + 3)² for every x ≠ −3.

Use the sign of f′(x) to explain why f is never decreasing.

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8Find2 marks

2026

For f(x) = x² − 10x + 3, the gradient function is f'(x) = 2x − 10. Find the values of x for which f is increasing.

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9Find2 marks

2026

For

f (x) = x^{2} - 10 x + 3

, the gradient function is

f^{'} (x) = 2 x - 10

. Find the values of x for which f is increasing.

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10Show that3 marks

2026

Show that

f (x) = x^{3} + 2 x

is increasing for all real x.

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11Show that3 marks

2026

Show that

f

is decreasing for

0 < x < 2

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12Find3 marks

2026

Find the interval on which

f

is decreasing.

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13State2 marks

2026

The graph of f' crosses the x-axis at x = −3 (going − to +) and at x = 1 (going + to −). State the nature of f at each x-value.

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14Find3 marks

2025

Let

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

. Find the values of

x

for which

f

is decreasing.

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15Find4 marks

2026

Consider a function f which has first derivative f′(x) = x² − 3x + 1, where x ∈ ℝ.

Find the range of values of x for which f is decreasing.

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16Find4 marks

2026

A function f satisfies f′(x) = x² + x − 3 for all real x.

Find the set of values of x for which f is decreasing.

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17Find2 marks

2026

For f(x) = 2x² − 12x + 7, find the value of x at which f stops decreasing and starts increasing.

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18Show that3 marks

2021

Let

f (x) = \frac{x}{x ^{2} + 4}

. Show that

f

is increasing for

- 2 < x < 2

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19Show that3 marks

2021

The function

g

is given by

g (x) = ln (x^{2} + 1) - x

. Show that the gradient of

g

is negative whenever

x > 1

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20Find4 marks

2026

A function

f

has derivative

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

. Find the values of

x

for which

f

is decreasing.

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