Page 45 - Focus SPM 2022 - Additional Mathematics

Additional Mathematics SPM Chapter 2 Differentiation
Determining the nature of the turning points: Solution
Tangent sketching method y = x + ax + b
3
At (3, 2): At (0, 4), 4 = (0) + a(0) + b
3
Choose one value Choose one value b = 4
where x , 3 where x . 3
2
x 2 3 4 Also, dy = 3x + a
dx
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dy 0 = 3(0) + a
2
dx –3 0 9 a = 0
Sign – zero + \ y = x + 4
3
Tangent dy 2
sketching dx = 3x
2
Graph and d y = 6x
sketching dx 2 = 6(0)

\ (3, 2) is a minimum point. = 0
\ The point (0, 4) is an inflection point.
At (1, 6):
x 0 1 2 Try Questions 6 – 8 in ‘Try This! 2.4’
dy
dx 9 0 –3 E Solving problems involving
maximum and minimum values and
Sign + zero –
interpreting the solutions
Tangent
sketching 1. Problems involving the maximum value and
minimum value can be solved by using the
Graph following steps:
sketching (a) Based on the information given in the
problem, recognise the main function
\ (1, 6) is a maximum point. (area, volume etc) that needs to be formed
or derived so that the differentiation can be
carried out. This main function can be easily
Alternative Method
recognised through the word “maximum”
Second derivative method or “minimum” stated in the problem.
d y = 6x – 12 (b) Obtain this main function and express it in
2
dx 2 terms of one variable. Normally, there would
x = 3, be one condition stated in the problem so that
d y = 6(3) – 12 the variables can be mutually substituted.
2
dy
Form 5
dx 2 = 6 . 0 (c) Let f'(x) or dx = 0 to obtain the value of
x = 1, the variable which causes the function to be
maximum or minimum.
d y = 6(1) – 12 (d) If there is more than one variables obtained,
2
dx 2 = –6 , 0 investigate the nature of the values by using
d y (second derivative method) or the
2
17 dx 2 dy
Given that the graph of the function y = x + ax + b, gradient of tangent dx .
3
where a and b are constants, has a turning point (0, 4). (e) Find the maximum value or the minimum
Determine the values of a and b. Hence, determine the value of the function according to the
nature of the turning point. requirement of the problem.

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