Mathematical Methods — Question 15

QCAA 2022, Paper 1 · 4 marks

Q15 · 2022 · Technology-freeComplex familiar4 marks

QUESTION 15 (4 marks)

The derivative of a function is given by

f'(x)=e^x(x-4)

. Determine the interval on which the graph of

f(x)

is both decreasing and concave up.

WORKED SOLUTION

4 marks

ANSWER

3<x<4

Worked solution

The function is decreasing when

f'(x)<0

and concave up when

f''(x)>0

. Since

f'(x)=(x-4)e^x<0\quad\text{when }x<4,

and

f''(x)=(x-4)e^x+e^x=e^x(x-3)>0\quad\text{when }x>3,

the function is decreasing and concave up when

3<x<4.

correctly describes conditions when the function is decreasing and concave up

[1 mark]

correctly determines the interval where

f(x)

is decreasing

[1 mark]

correctly determines the interval where

f(x)

is concave up

[1 mark]

determines interval when function is decreasing and concave up

[1 mark]

One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.

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