Cambridge International AS and A Level Mathematics Pure Mathematics 1

Differentiation

P1^

5

This is an example of a point of inflection. In general, a point of inflection occurs

where the tangent to a curve crosses the curve. This can happen also when d

d

y

x

≠ 0,

as shown in figure 5.24.

Figure 5.24

Notice that the

gradient of the

curve on either side

of the point has the

same sign.

Points of inflection

If you are a driver you may find it helpful to think of a point of inflection as the

point at which you change from left lock to right lock, or vice versa. Another way

of thinking about a point of inflection is to view the curve from one side and see

it as the point where the curve changes from being concave to convex.

The second derivative

Figure 5.25 shows a sketch of a function y = f(x), and beneath it a sketch of the

corresponding gradient function d

d

y

x

= f ́(x).

y

O

d––y

dx

O

x

x

P

Q

Figure 5.25