Page 44 - Focus SPM 2022 - Additional Mathematics

Additional Mathematics SPM Chapter 2 Differentiation
(a) Tangent sketching method
D Determining the turning points and
their nature
(i) Maximum
1. Turning points or stationary points, are the
points on a curve where the gradient of the
tangent at the points are 0. The points have either (ii) Minimum
maximum or minimum value.
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(a) Maximum point (iii) Inflection or
point
y
0
positive negative (b) Second derivative method
2
x d y
O (i) Maximum , 0
dx 2
The value of the gradient of the tangent (ii) Minimum d y . 0
2
changes from positive to zero to negative, dx 2
from left to right.
(iii) Inflection d y = 0
2
(b) Minimum point point dx 2
y
16
negative positive Find the turning points of the function y = x – 6x + 9x + 2
3
2
0 and determine the nature of each turning point.
x
O
Solution
The value of the gradient of the tangent y = x – 6x + 9x + 2
3
2
changes from negative to zero to positive,
from left to right. dy = 3x – 12x + 9
2
dx
2. If the value of the stationary point is neither At the turning point,
maximum nor minimum, it is called an 0 = 3x – 12x + 9
2
inflection point. x – 4x + 3 = 0
2
y y (x – 3)(x – 1) = 0
negative
positive x = 3, x = 1
x x
O O When x = 3,
positive negative Form 5
y = (3) – 6(3) + 9(3) + 2
3
2
The value of the gradient of the tangent changes = 27 – 54 + 27 + 2
from positive or negative to zero to positive or = 2
negative back in both directions.
When x = 1,
3. To determine the turning point, let dy = 0. y = (1) – 6(1) + 9(1) + 2
3
2
dx
= 1 – 6 + 9 + 2
4. To determine the nature of the turning point,
either maximum, minimum or inflection point, = 6
the following two methods can be applied: \ The turning points are (3, 2) and (1, 6).

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