Let R be the reals. Let f : R → R have a continuous third derivative. Show that there is a point a with f(a) f '(a) f ''(a) f '''(a) ≥ 0.
Solution
Straightforward.
If any of f, f ', f '', f ''' change sign, then we are done. We show that if f, f ' and f '' do not change sign, then f and f '' have the same sign. Suppose f '' is positive. Then f(x) = f(0) + f '(0) x + f ''(c) x2/2 for some c, so for all x, f(x) > f(0) + f '(0) x. That is certainly positive for x sufficiently large and of the same sign as f '(0). Hence it is positive for all x. Similarly, if f '' is negative, then f is negative. Hence f(x)f ''(x) ≥ 0 for all x. Similarly f '(x)f '''(x) ≥ 0 for all x.
© John Scholes
jscholes@kalva.demon.co.uk
12 Dec 1998