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Suppose $ f $ is continuous on $ [1, 5] $ and the only solutions of the equation $ f(x) = 6 $ are $ x = 1 $ and $ x = 4 $. If $ f(2) = 8$, explain why $ f(3) > 6 $.

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03:17

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 5

Continuity

Limits

Derivatives

Leo L.

January 16, 2022

How does, "only solutions of f(x) = 6 are x=1 and x=4," imply that you "cannot cross the line"? Since the function is not given, how do I know that f(2) is not 1 or some other number less than 6?

Am I missing some Axiom?

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Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

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Alright, so here's an interesting question. They don't tell you what the function is that, they tell you it's continuous on the interval from 1-5 inclusive. And then this is an interesting thing. F uh The only solutions to f of X equals six Are one and 4. Let's start graphing. This is okay, so we've got six, maybe that's three. I'm just trying to see how big one should be. Okay, one and four, So we've got at X equals one, and X equals four. The value is six, and then I'm just going to put a dotted line here at no other point. Are we allowed to cross this dotted line? That's what this only means. The only time we're allowed to touch that line, Is that one and 4? Okay. Furthermore, we have F two equals eight. We have this point on the function and it's got to be continuous All the way from 1 to 5 in that hole in that whole interval. Okay. Um So now we're asking, well, what's the value at 3? Well um it's got to be continuous. So somehow either, you know, it can be as crazy as you want, but we've got to end up here and then we've got to end up here and we've got to end up Back down to six without crossing this line. So there is no way that the value at three could be below six and be continuous. If it were we would across this line and we haven't. Um Okay, and the fancy way of saying this is the intermediate value theorem. Uh Well, is it the no, this is not really the intermediate value, the intermediate value theorem is related, but basically we've got to be continuous. We can't cross this line, so um three has to be greater than six.

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