# Bolzano's Theorem

If f is continuous in *[a,b]* and *f(a)·f(b) < 0*, then

NOTE: this theorem is a tool to approximate a root of an unsolvable equation or to show that it exists.

Example: demonstrate that the equation *x ^{3} – 3x + 40 = 0* has a real root and approximate it to the tenths.

Let *f(x) = x ^{3} – 3x + 40 *

*f* is continuous in **R**, because it is a polynomial function, and *f(-4) = -12*, *f(-3) = 22,* then, using the Bolzano’s Theorem, *c* exists in (*-4,-3)* such that *f(c) = 0*

↔ *c ^{3} – 3c + 40 = 0*↔

*c*is a root of the equatio

*n.*

* *In addition* f(-3,8) = -3,472 and* *f(-3,7) = 0,447* then, using again the Theorem, *c ≈-3,7*

* *

**Exercises**

1.- (PAEG- june 2009): Demonstrate that the graphs of the functions:

intersect at least in one point.

2.- The function

is defined in the interval [0,1] and *f(0)·f(1)<0*, but doesn't exist any *c* in [0,1] such that *f(c) = 0*. Does it contradict Bolzano's Theorem?

Solutions: 1.- Use Bolzano's Theorem for the function *f - g*; 2.- It doesn't contradict Bolzano's Theorem because *f* is not continuous in [0,1]

* *

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