CIE/GCE/AS/Math/P1/20/Mar/12/Q#1
Question
1
The function f is defined by 𝑓 (𝑥 ) = + 𝑥 2 for 𝑥 < −1.
3𝑥+2
Determine whether f is an increasing function, a decreasing function or neither.
Solution
We are given function;
1
𝑓 (𝑥 ) = + 𝑥2
3𝑥+2
We are required to find whether 𝑓(𝑥 ) is an increasing function, decreasing function or
neither.
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
𝑑𝑦
Let’s find of the given function.
𝑑𝑥
𝑑𝑦 𝑑 1
= ( + 𝑥2 )
𝑑𝑥 𝑑𝑥 3𝑥 + 2
𝑑𝑦 𝑑
= ( (3𝑥 + 2)−1 + 𝑥 2 )
𝑑𝑥 𝑑𝑥
Rule for differentiation of 𝑦 = 𝑔(𝑥 ) + ℎ(𝑥) is:
𝑑 𝑑 𝑑
[𝑔(𝑥 ) + ℎ(𝑥)] = [𝑔(𝑥)] + [ℎ(𝑥)]
𝑑𝑥 𝑑𝑥 𝑑𝑥
Question
1
The function f is defined by 𝑓 (𝑥 ) = + 𝑥 2 for 𝑥 < −1.
3𝑥+2
Determine whether f is an increasing function, a decreasing function or neither.
Solution
We are given function;
1
𝑓 (𝑥 ) = + 𝑥2
3𝑥+2
We are required to find whether 𝑓(𝑥 ) is an increasing function, decreasing function or
neither.
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
𝑑𝑦
Let’s find of the given function.
𝑑𝑥
𝑑𝑦 𝑑 1
= ( + 𝑥2 )
𝑑𝑥 𝑑𝑥 3𝑥 + 2
𝑑𝑦 𝑑
= ( (3𝑥 + 2)−1 + 𝑥 2 )
𝑑𝑥 𝑑𝑥
Rule for differentiation of 𝑦 = 𝑔(𝑥 ) + ℎ(𝑥) is:
𝑑 𝑑 𝑑
[𝑔(𝑥 ) + ℎ(𝑥)] = [𝑔(𝑥)] + [ℎ(𝑥)]
𝑑𝑥 𝑑𝑥 𝑑𝑥
