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Date November 2021 Marks available 3 Reference code 21N.1.SL.TZ0.9
Level Standard Level Paper Paper 1 (without calculator) Time zone Time zone 0
Command term Justify and Find Question number 9 Adapted from N/A

Question

Consider a function f with domain a<x<b. The following diagram shows the graph of f', the derivative of f.

The graph of f', the derivative of f, has x-intercepts at x=p, x=0 and x=t . There are local maximum points at x=q and x=t and a local minimum point at x=r.

Find all the values of x where the graph of f is increasing. Justify your answer.

[2]
a.

Find the value of x where the graph of f has a local maximum.

[1]
b.

Find the value of x where the graph of f has a local minimum. Justify your answer.

[2]
c.i.

Find the values of x where the graph of f has points of inflexion. Justify your answer.

[3]
c.ii.

The total area of the region enclosed by the graph of f', the derivative of f, and the x-axis is 20.

Given that fp+ft=4, find the value of f0.

[6]
d.

Markscheme

Special note: In this question if candidates use the word 'gradient' in their reasoning. e.g. gradient is positive, it must be clear whether this is the gradient of f or the gradient of f' to earn the R mark.

 

f increases when p<x<0                A1

f increases when f'x>0   OR   f' is above the x-axis                R1


Note: Do not award A0R1.

 

[2 marks]

a.

Special note: In this question if candidates use the word 'gradient' in their reasoning. e.g. gradient is positive, it must be clear whether this is the gradient of f or the gradient of f' to earn the R mark.

 

x=0               A1

 

[1 mark]

b.

Special note: In this question if candidates use the word 'gradient' in their reasoning. e.g. gradient is positive, it must be clear whether this is the gradient of f or the gradient of f' to earn the R mark.

 

f is minimum when x=p               A1

because f'p=0, f'x<0 when x<p and f'x>0 when x>p

(may be seen in a sign diagram clearly labelled as f')

OR because f' changes from negative to positive at x=p

OR f'p=0 and slope of f' is positive at x=p               R1

 

Note: Do not award A0 R1

 

[2 marks]

c.i.

Special note: In this question if candidates use the word 'gradient' in their reasoning. e.g. gradient is positive, it must be clear whether this is the gradient of f or the gradient of f' to earn the R mark.

 

f has points of inflexion when x=q, x=r and x=t               A2


f' has turning points at x=q, x=r and x=t

OR

f''q=0, f''r=0 and f''t=0 and f' changes from increasing to decreasing or vice versa at each of these x-values (may be seen in a sign diagram clearly labelled as f'' and f')              R1

 

Note: Award A0 if any incorrect answers are given. Do not award A0R1

 

[3 marks]

c.ii.

Special note: In this question if candidates use the word 'gradient' in their reasoning. e.g. gradient is positive, it must be clear whether this is the gradient of f or the gradient of f' to earn the R mark.

 

recognizing area from p to t (seen anywhere)               M1

ptf'xdx

recognizing to negate integral for area below x-axis               (M1)

p0f'xdx-0tf'xdx   OR   p0f'xdx+t0f'xdx

mnf'xdx=fn-fm (for any integral)               (M1)

f0-fp-ft-f0   OR   f0-fp+f0-ft               (A1)

2f0-ft+fp=20,  2f0-4=20               (A1)

f0=12               A1

 

[6 marks]

d.

Examiners report

[N/A]
a.
[N/A]
b.
[N/A]
c.i.
[N/A]
c.ii.
[N/A]
d.

Syllabus sections

Topic 5 —Calculus » SL 5.2—Increasing and decreasing functions
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Topic 5 —Calculus

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