I'm doing a review packet for Calculus and I'm not really sure what it is...

Montenovofe

Montenovofe

Answered

2022-07-10

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?

Answer & Explanation

Jamiya Costa

Jamiya Costa

Expert

2022-07-11Added 18 answers

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 [ 3 , 6 ] such that f ( x ) = c for any c [ 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 ( 3 , 6 ). I hope this helps make the solution a little bit more clear.
bandikizaui

bandikizaui

Expert

2022-07-12Added 7 answers

It means that for every value c [ f ( 3 ) , f ( 6 ) ] = [ 2 , 3 ] there exists at least one value x c [ 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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