CIE/GCE/AS/Math/P1/19/Nov/13/Q#8
Question
1
1
A function 𝑓 is defined for 𝑥 > and is such that 𝑓 ′ (𝑥) = 3(2𝑥 − 1)2 − 6.
2
i. Find the set of values of 𝑥 for which f is decreasing.
ii. It is now given that 𝑓 (−1) = −3. Find 𝑓(𝑥 ).
Solution
i.
We are given derivative of the function as;
1
𝑓 ′ (𝑥) = 3(2𝑥 − 1)2 − 6
We are also given that it is a decreasing function.
To test whether a function 𝑦 = 𝑓(𝑥) is increasing or decreasing at a particular point
𝑃(𝑥𝑃 , 𝑦𝑃 ), we take derivative of a function at that point.
𝑑𝑦
If | > 0, the function 𝑦 = 𝑓(𝑥) is increasing.
𝑑𝑥 𝑥
𝑃
𝑑𝑦
If | < 0 , the function 𝑦 = 𝑓(𝑥) is decreasing.
𝑑𝑥 𝑥
𝑃
𝑑𝑦
If | = 0 , the test is inconclusive.
𝑑𝑥 𝑥
𝑃
Since we are given that function is decreasing;
𝑑𝑦
<0
𝑑𝑥
Therefore;
1
3(2𝑥 − 1)2 − 6 < 0
Question
1
1
A function 𝑓 is defined for 𝑥 > and is such that 𝑓 ′ (𝑥) = 3(2𝑥 − 1)2 − 6.
2
i. Find the set of values of 𝑥 for which f is decreasing.
ii. It is now given that 𝑓 (−1) = −3. Find 𝑓(𝑥 ).
Solution
i.
We are given derivative of the function as;
1
𝑓 ′ (𝑥) = 3(2𝑥 − 1)2 − 6
We are also given that it is a decreasing function.
To test whether a function 𝑦 = 𝑓(𝑥) is increasing or decreasing at a particular point
𝑃(𝑥𝑃 , 𝑦𝑃 ), we take derivative of a function at that point.
𝑑𝑦
If | > 0, the function 𝑦 = 𝑓(𝑥) is increasing.
𝑑𝑥 𝑥
𝑃
𝑑𝑦
If | < 0 , the function 𝑦 = 𝑓(𝑥) is decreasing.
𝑑𝑥 𝑥
𝑃
𝑑𝑦
If | = 0 , the test is inconclusive.
𝑑𝑥 𝑥
𝑃
Since we are given that function is decreasing;
𝑑𝑦
<0
𝑑𝑥
Therefore;
1
3(2𝑥 − 1)2 − 6 < 0
