Cambridge International AS and A Level Mathematics Pure Mathematics 1

P1^

5

Points of inflection

153

13  (i) Find the co-ordinates of the stationary points of the curve f(x) = 4x +^1

x

.

(ii) Find the set of values of x for which f(x) is an increasing function.

14  The equation of a curve is y = 16 (2x − 3)^3 − 4 x.

(i) Find d

d

y

x

.

(ii) Find the equation of the tangent to the curve at the point where the curve

intersects the y axis.

(iii) Find the set of values of x for which^16 (2x − 3)^3 − 4 x is an increasing

function of x.

[Cambridge AS & A Level Mathematics 9709, Paper 12 Q10 June 2010]

15  The equation of a curve is y = x^2 – 3x + 4.

(i) Show that the whole of the curve lies above the x axis.

(ii) Find the set of values of x for which x^2 – 3x + 4 is a decreasing function of x.

The equation of a line is y + 2x = k, where k is a constant.

(iii) In the case where k = 6, find the co-ordinates of the points of intersection

of the line and the curve.

(iv) Find the value of k for which the line is a tangent to the curve.

[Cambridge AS & A Level Mathematics 9709, Paper 1 Q10 June 2005]

16  The equation of a curve C is y = 2x^2 – 8x + 9 and the equation of a line L is

x + y = 3.

(i) Find the x co-ordinates of the points of intersection of L and C.

(ii) Show that one of these points is also the stationary point of C.

[Cambridge AS & A Level Mathematics 9709, Paper 1 Q4 June 2008]

Points of inflection

It is possible for the value of d

d

y

x

to be zero at a point on a curve without it being a

maximum or minimum. This is the case

with the curve y = x^3 , at the point (0, 0)

(see figure 5.23).

y = x^3 ⇒ d

d

y

x

= 3 x^2

and when x = 0, d

d

y

x

= 0. x

y

O

Figure 5.23