Calculus: Analytic Geometry and Calculus, with Vectors

5.3 Second derivatives, convexity, andfexpoints 307

calculate f'(x) and f"(x). Show that the graph of f has exactly one flexpoint
for which x = -b/3a and that f is increasing when x > -b/3a.
5 Show that if the x, y coordinate system is chosen in such a way that the
graph of

y = x3+bx2+cx+d

passes through the origin and has its flexpoint at the origin, then b = d = 0 and

Y = x3 + cx.

Shoe that the graph of the latter equation has no e'trema if c > 0 and has two
local e'trema if c < 0.
6 Starting with the first of the relations

(1) y = (a2 - x2);6f

d2y a2

dx2 - y3'

differentiate twice and obtain the second relation. Then start with the first

of the relations

(2)

(3)

x2+y2=a2, x+yda=0

1+''ax +(ax) =0

and show that differentiating with respect to x gives the others. Use (2) and

(3) to obtain the second relation in (1). Tell why you should expect the sign of

the second derivative to be opposite to the sign of y.

7 Supposing that a and p are given positive numbers and considering posi-

tive values of x and y, use the two methods of the preceding problem to find

d2y/dx2 when

xp + yv = aP.

Make the results agree with each other and, for the case p = 2, with a result of

the preceding problem. Tell why the sign of the second derivative should (or

should not) depend upon p as it does in your answer.

8 Supposing that a > 0 and b > 0, show that the graph of

f(x) = a sin (bx + c)

has a flexpoint wherever it intersects the x axis.

9 Sketch a reasonably accurate graph of the function f for which

f(x) = x sin x

and observe that the graph seems to have flexpoints on or near the x axis.

Show that if (x,y) is a flexpoint, then tan x = 2/x and y = 2 cos x. Remark:

These results show that if (x,y) is a flexpoint for which IxI is large, then tan x is

near 0, sin x is near 0, cos x is near 1 or -1, and y is near 2 or -2.

10 Supposing that is is 10 or 20, sketch the graph of y = sins x over the

interval 0 5 x 5 7r/2 and mark a point which seems to be a flexpoint. Then,