Let f(x)=3x^5+5x^3+15x+15.

Question

Use Rolle's Theorem to show that f(x) has exactly one root.

Answer 1

f' = 15 x^4 + 15 x^2 + 15
= 15 (x^4+x^2+1)
where is that slope = 0?
x^2 = [ -1 +/- sqrt(1-4)]/ 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.