Question 1
Calculate the derivatives of the following functions:
(a) ??(??) =
??
2
??+1
(b) ??(??) = ??
3
· ??
2??
(c) ??(??) = ln(?? + v??
2 + 1)
(d) ??(??) = v1 + v1 + ??
[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).
v2 + ?? <
?? + 18
8
, for ?? > 14
[30 marks]