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I'm doing a review packet for Calculus and I'm not really sure what it is asking for the answer?
The question is: Let f be a continuous function on the closed interval [-3, 6]. If f(-3)=-2 and f(6)=3, what does the Intermediate Value Theorem guarantee? I get that the intermediate value theorem basically means but not really sure how to explain it?

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3 Answers 3

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Since $f(-3)=-2<0<3=f(6)$, we can guarantee that the function has a zero in the interval $[-3,6]$. We cannot conclude it has only one, though (it may be many zeros).

EDIT: As has already been pointed out elsewhere, the IVT guarantees the existence of at least one $x\in[-3,6]$ such that $f(x)=c$ for any $c\in[-2,3]$. Note that the fact that there is a zero may be important (for example, you couldn't define a rational function over this domain with this particular function in the denominator), or you may be more interested in the fact that it attains the value $y=1$ for some $x\in(-3,6)$. I hope this helps make the solution a little bit more clear.

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While the zeroes are usually the most important points we might wish to find, the IVT also guarantees every other value between $f(a)$ and $f(b)$ occurs at least once between $a$ and $b$. –  Todd Wilcox Jan 24 '13 at 3:09

It means that for every value $\,c\in [f(-3),f(6)]=[-2,3]\,$ there exists at least one value $\,x_c\in [-3,6]\,$ s.t. $\,f(x_c)=c\,$.

The above, for example, tells us the function has a zero in $\,[-3,6]\,$...

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Note that the continuity of $f$ is absolutely crucial to the Intermediate Value Theorem. It might be instructive to ponder for a moment why this is. –  Austin Mohr Jan 24 '13 at 3:00

The actual statement of the Intermediate Value Theorem is as follows.

Let $f$ be a continuous function on the closed interval $[a,b].$ Then for any $c \in (f(a),f(b))$ (this is an open interval), there exists $x \in (a,b)$ such that $f(x) = c.$

In your case, it guarantees the existence of a zero in the interval $[-3,6].$

Something interesting to note is that if $f$ is differentiable on $[a,b],$ then $f^{\prime}$ has the intermediate value property. This is known as Darboux's Theorem and is a nice exercise.

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