No particular specialpointsspring to mind, except thestationary
points and points of inflection. We know from Review
Question 10.1.3(i) that there are two stationary points, atx=±^43.
The point atx=^43 is a maximum and that atx=−^43 is a minimum.
There is also a point of inflection atx=0 (this follows by symmetry
anyway). This enables us to complete the sketch as shown in
Figure 10.7.y = 16 x − 3 x^3y0 4 x
34
√ 34
34
√ 3− −Figure 10.7Sketch ofy= 16 x− 3 x^3.(ii) y=x+
1
x
Again, this is an odd function and so is symmetric under a 180°
rotation about the origin and we need only sketch it forx≥0.
Gateways: The curve does not cross the y-axis, since there is a
discontinuity atx=0. Also, it never crosses thex-axis, sincex+1
x=x^2 + 1
x
can never be zero.
Restrictions:Forx>0,y>0, and forx<0,y<0, so the curve
is confined to the upper half plane forx>0 and to the lower half
plane forx<0.
Asymptotes:Asx→0 from above (that is through positive values
ofx),y→+∞,sothey-axis is a vertical asymptote forx>0. Also,
forxvery large,y∼x, so the straight liney=xis an asymptote as
x→∞. Consideration of the asymptotes alone hints at a minimum
forx>0 (and a corresponding maximum forx<0).
Special points:nothing, apart from:
Stationary points: From Review Question 10.1.3(ii) we have a
minimum atx=1, and a maximum atx=−1.
Putting all this together, the graph is as shown in Figure 10.8.