Elgin Johnston
Problems Editor
Department of Mathematics
Iowa State University
Ames, IA 50011
1668. Proposed by Steve Butler, Provo, UT
Let ƒ be a real valued function defined on an open interval I containing [a, b].
Assume that ƒ has a continuous second derivative on I and that there is a single line
tangent to the graph of y = ƒ(x) at (a, ƒ(a)) and (b, ƒ(b)). Prove that if ƒ’’(x) is not
identically zero on (a, b), then ƒ’’(x) must change sign at least twice on (a,b)
Proof.
1.)
Let Ts(x) be the equation of the line tangent to ƒ(x) at (s, ƒ(s)); then
Ta(x) = [ƒ’(a)]x + r, for some constant r.
Define h(x) = ƒ(x) – Ta(x) . Then, h’(a) = h’(b) = 0 and h’’(x)  ƒ’’(x) on I.
2.)
From the hypotheses, h(a) = h(b) = 0.
f’’(x), and consequently h’’(x), is not identically zero; the graph of h(x) is not a
>
line on [a, b] and hence for some t  (a, b), h(t) < 0.
3.)
Suppose h(t) > 0. Then h(x) assumes a positive maximum at c  (a, b).
h(a) = 0; h(c) > 0; by MVT, d  (a, c) such that h’(d) > 0.
By construction, h’(a) = 0; h’(d) > 0; by MVT, e  (a, d) such that ƒ’’(e) > 0.
h’(d) > 0; h’(c) = 0, since h(c) is an extremum; by MVT   (d, c) such that
h’’() < 0.
h’’(x) is continuous, so h’’(x) changes sign at least once on (a, c).
h’’(x)  ƒ’’(x) on I, so ƒ’’(x) changes sign at least once on (a, c). Similarly, ƒ’’(x)
changes sign at least once on (c, b).
5.)
If h(t) < 0, the same conclusion follows.
QED