Posted by Joe on Sunday, November 13, 2011 at 5:49pm.
Let f(x)=3x^5+5x^3+15x+15.
Use Rolle's Theorem to show that f(x) has exactly one root.

Calculus  Damon, Sunday, November 13, 2011 at 5:58pm
f' = 15 x^4 + 15 x^2 + 15
= 15 (x^4+x^2+1)
where is that slope = 0?
x^2 = [ 1 +/ sqrt(14)]/ 2
x^2 = 1/2 +/ (1/2) sqrt (3)
complex roots only, it never has zero slope so it can only cross the axis once by Rolle's theorem. Once it crosses the axis, it can never reverse and come back.