4
Leg A: Let f (x) = −5√𝑥 3 . Find f (x).
Working:
( 6 marks)
Leg B:
Differentiate with respect to x
(a) √6 − 2𝑥
(b) 𝑒 ln 𝑥
Working:
(4 marks)
Leg C:
Let g(x) = 3x cos x.
(a) Find g′(x).
(b) Find the gradient of the graph of g at x = π.
(7 marks)
Leg D:
Let ℎ(𝑥 ) =
ln 𝑥
𝑥3
, for x > 0.
(a) Use the quotient rule to find h’(x).
(b) The graph of g has a maximum point at A. Find the xcoordinate of A.
(7 marks)
4
Leg E: Let f(x) = kx . The point P(1, k) lies on the curve of f.
At P, the normal to the curve is parallel to y =

1
x.
8
Find the
value of k.
(Total 6 marks)
Leg F:
Let g(x) = 3x4 – 16x3 + 24x2 – 2.
(a) Find the two values of x at which the tangent to the graph
of g is horizontal.
(b) For each of these values, determine whether it is a
maximum or a minimum.
(14 marks)
Leg G: Let f′(x) = –24x3 + 9x2 + 3x + 1.
(a) There are two points of inflexion on the graph of f. Write
down the x-coordinates of these points.
(b) Let g(x) = f″(x). Explain why the graph of g has no points
of inflexion.
(Total 5 marks)
Leg H: Consider the function f (x) = 4x3 + 2x. Find the equation
of the normal to the curve of f at the point where x =1.
(Total 6 marks)
Leg I: The diagram shows part of the
graph of y = f′(x). The x-intercepts are
at points A and C.There is a minimum
at B, and a maximum at D.
(a) (i) Write down the value
of f′(x) at C.
(ii) Hence, show that C
corresponds to a minimum
on the graph of f, i.e. it has
the same x-coordinate.
(b) Which of the points A, B, D corresponds to a maximum on the
graph of f?
(c) Show that B corresponds to a point of inflexion on the graph of f.
(7 marks)
Leg J: A curve has equation y = x(x – 4)2.
(a) For this curve find
(i) the x-intercepts;
(ii) the coordinates of the maximum point;
(iii) the coordinates of the point of inflexion.
(9)
(b) Use your answers to part (a) to sketch a graph of the curve
for 0  x  4, clearly indicating the features you have
found in part (a).
(3)