Mathematical Methods — Question 13

QCAA 2020, Paper 1 · 7 marks

Q13 · 2020 · Technology-freeSimple familiar7 marks

QUESTION 13 (7 marks)

A function is defined as

f(x)=x(\ln(x))^2

x>0

. The graph of the function is shown and has a local maximum at point A and a global minimum at point B. The derivative of the function is given by

f'(x)=2\ln(x)+(\ln(x))^2

x>0

Graph of f(x)=x(ln x)^2 showing local maximum A and global minimum B.

Verify that there is a stationary point at

x=1

[2 marks]

Determine the coordinates of A.

[3 marks]

The graph of the function has a point of inflection at

x=e^p

. Determine

p

[2 marks]

Question linkOriginal paper

WORKED SOLUTION

QCAA guide · typeset solution

7 marks

ANSWER

a) stationary at

x=1

; b)

A(e^{-2},4e^{-2})

; c)

p=-1

Worked solution

a) Set

f'(x)=0

0=2\ln x+(\ln x)^2=\ln x(2+\ln x).

One solution is

\ln x=0

, hence

x=1

b) The other stationary point satisfies

2+\ln x=0

, so

x=e^{-2}

. Then

y=e^{-2}(\ln e^{-2})^2=4e^{-2}

. Thus

A(e^{-2},4e^{-2})

f''(x)=\frac{2}{x}+\frac{2\ln x}{x}=\frac{2}{x}(1+\ln x).

Setting

f''(x)=0

gives

x=e^{-1}

, so

p=-1

correctly identifies that the derivative equals 0

[1 mark]

correctly shows there is a stationary point at

x=1

[1 mark]

correctly establishes the equation in

x

[1 mark]

correctly determines the

x

-ordinate of A

[1 mark]

determines

y

-ordinate of A

[1 mark]

correctly establishes the equation in

x

equals 0

[1 mark]

determines

p

[1 mark]

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Mathematical Methods — Question 13

QUESTION 13 (7 marks)

QCAA guide · typeset solution

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