# 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.

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### 1 Answer

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.**