On-line Math 21

4.1  The Mean Value Theorem

Example 3 Show that the function f(x) = x⁵+4x³+3x-8 has only one root, one point x where f(x) = 0 .

Solution

If f(x) had two points, call them a and b , a ¹ b , with f(a) = 0 and f(b) = 0 , then there would have to be, by the MVT (actually, by Rolle's Theorem) a point c between a and b for which f¢(c) = 0 . However,

f¢(x) = 5x4+4x2+3,

so no matter what c is, f¢(c) ¹ 0 , in fact, f¢(c) ³ 3 at every point. Since this can't exist, the assumption, that there were two distinct roots of the equation, had to have been wrong.

Copyright (c) 2000 by David L. Johnson.