Using the Intermediate Value Theorem to Determine the Existence of a Zero Crossing in a Continuous Function over an Interval

If f(1)4 and f(6)=9, then there must be a x-value between 1 and 6 where f crosses the x-axis.

Intermediate Value Theorem

We cannot make that conclusion directly based on the values given for f(1) and f(6). However, if we assume that f is a continuous function over the interval [1,6], then we can apply the Intermediate Value Theorem.

According to the Intermediate Value Theorem: if a function f(x) is continuous on the closed interval [a,b], and if c is any number between f(a) and f(b), then there must be at least one number x in the interval (a, b) such that f(x) = c.

In this case, we know that f(1) = 4 and f(6) = 9. Since f is assumed to be continuous over [1,6], any number between 4 and 9 must be achieved by f at some point between x=1 and x=6. In particular, the value 0 (which corresponds to crossing the x-axis) is between 4 and 9, so there must be at least one value of x between 1 and 6 where f(x) = 0.

Therefore, based on the Intermediate Value Theorem, we can conclude that there is likely an x-value between 1 and 6 where f crosses the x-axis. However, without additional information about the behavior of f, we cannot definitively determine the exact location of this x-value.

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