Regarding the Bolzzano Principle, explain how, given function f with domain [0,1] and f(x)=6x^3+4x-9 has a value a in (0,1) so that f(a)=0, without solving the equations.
Bolzano's Theorem, more commonly known as the Intermediate Value Theorem, tells us that given two points on a continuous graph, where one point is above some horizontal line and one point is below the same horizontal line, there must be a point between the two given points that is on said line.
For your problem, since you are looking for f(a) = 0, the horizontal line of interest is the x-axis. Therefore, you need to show that in the given interval [0, 1], you have a point above the x-axis and a point below the x-axis. To do so, calculate f(0) and f(1). You should find that f(0) = -9 and f(1) = 1. Since one value is negative (so the point is below the x-axis) and one is positive (above the x-axis), there must be some value a in [0, 1] such that f(a) = 0.