Introduction to integration — Question 120

Original QCE Vault practice · 5 marks

Q120 · Practice questionTechnology-freeComplex unfamiliar5 marks

QUESTION 120 (5 marks)

A twice-differentiable function satisfies

f''(x)=6x-4

f'(1)=0

and

\displaystyle\int_0^2 f(x)\,dx=10

. Determine

f(0)

WORKED SOLUTION

5 marks

ANSWER

f(0)=\dfrac{14}{3}

Worked solution

Integrating once gives

f'(x)=3x^2-4x+C

. From

f'(1)=0

C=1

. Hence

f(x)=x^3-2x^2+x+D

. Integrating from

0

2

gives

2/3+2D=10

, so

D=14/3=f(0)

Integrates

f''

correctly.

[1 mark]

Uses

f'(1)=0

to find the first constant.

[1 mark]

Integrates to obtain

f(x)

[1 mark]

Uses the definite-integral condition correctly.

[1 mark]

Obtains

f(0)=14/3

[1 mark]

Practice question aligned to the current QCAA syllabus; review the worked solution and marking criteria.

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