Understanding Engineering Mathematics

In this casef′′(x)is never zero for finitex, so we have no points

of inflection. Notice that in this case the local minimum value (2) is

greater than the local maximum value (−2), emphasising the local

nature of differentiation. The situation is sketched in Figure 10.8.

(iii) Forf(x)=xexwe havef′(x)=ex+xex=0whenx=−1, since
ex=0. At this stationary point we have

f′′(x)= 2 ex+xex=e−^1 >0atx=− 1

Sox=−1isaminimum pointin this case.

Looking for points of inflection we note thatf′′(x)=ex(x+ 2 )=0at

x=−2. Forx<−2,f′′(x) <0,f′(x)decreases asxincreases and

the curve is concave down. Forx>−2,f′′(x) >0,f′(x)increases

asxincreases and the curve is concave up. Sox=−2 is a point of

inflection. The curve is sketched in Figure 10.9.

(iv) We can writef(x)=sinxcosx=^12 sin 2x.

Thenf′(x)=cos 2x=0when2x=

( 2 n+ 1 )

2

πwherenis an integer.

So there are turning points where

x=

(

2 n+ 1

4

)

π

Further,

f′′(x)=−2sin2x=−2sin

((

2 n+ 1

2

)

π

)

Forneven this is negative, while it is positive fornodd.

Thus we have maximum when x=

(

2 n+ 1

4

)

π forn even, and

minimum whennis odd.

Points of inflection can occur whenf′′(x)=−2sin2x=0, i.e.

2 x=mπ

wheremis an integer, orx=

mπ

2

.

(v) Forf(x)=x^4 we havef′(x)= 4 x^3 =0whenx=0. So there is a

stationary point atx=0. But

f′′(x)= 12 x^2 =0whenx= 0

So there may be a point of inflection atx=0. However, we note that

forx<0,f′′(x) >0, so the curve is concave up, while forx>0we

again havef′′(x) >0, again indicating that the curve is concave up.

So the concavity does not change atx=0 and therefore it isnota point

of inflection. In fact, the graph is easy to sketch, and we clearly have a

minimum atx=0 (see Figure 10.11, Review Question 10.1.4(v). This

illustrates that the vanishing of the second derivative, while essential

for a point of inflection is not a guarantee of one.