Question 1
Calculate the derivatives of the following functions:
(a) ?(?) =
?
2
?+1
(b) ?(?) = ?
3
∙ ?
2?
(c) ?(?) = ln(? + √?
2 + 1)
(d) ?(?) = √1 + √1 + ?
[40 marks]
Question 2
Consider the function ?(?) = ?
3 + ?? , where ? is a real parameter. Find its stationary
point(s) and discuss the nature of the stationary point(s), in the case where ? > 0, ? = 0 and ?
< 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:
?
′
(?) =
?(?) − ?(?)
? − ?
Use the Mean Value Theorem to establish the following inequality (assuming any relevant
derivative formulas).
√2 + ? <
? + 18
8
, for ? > 14
[30 marks]