Question 1
Calculate the derivatives of the following functions:
(a) 𝑓(𝑥) =
𝑥
2
𝑥+1
(b) 𝑓(𝑥) = 𝑥
3
∙ 𝑒
2𝑥
(c) 𝑓(𝑥) = ln(𝑥 + √𝑥
2 + 1)
(d) 𝑓(𝑥) = √1 + √1 + 𝑥
[40 marks]
Question 2
Consider the function 𝑓(𝑥) = 𝑥
3 + 𝜆𝑥 , where 𝜆 is a real parameter. Find its stationary
point(s) and discuss the nature of the stationary point(s), in the case where 𝜆 > 0, 𝜆 = 0 and 𝜆
< 0.
[30 marks]
Question 3
The Mean Value Theorem states the following: Let f be a continuous function on [a, b] that
is differentiable on (a, b). Then there exists at least one point c ∈ (a, b) such that:
𝑓
′
(𝑐) =
𝑓(𝑏) − 𝑓(𝑎)
𝑏 − 𝑎
Use the Mean Value Theorem to establish the following inequality (assuming any relevant
derivative formulas).
√2 + 𝑥 <
𝑥 + 18
8
, for 𝑥 > 14
[30 marks]
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