Let be a continuous function such that and .
Using the Intermediate Value Theorem classify the following statements as:
( A ) Always true
( B ) Never True, or
( C ) True in some cases; False in others.
1. f(0)=0
(answer a,b, or c) _______
2. For some c , where , .
(answer a,b, or c) _______
thank you
chad
I have the intermediate value theorem right here in front of me and i have read over it multiple times but i guess i dont get it.
for number 1 i thought it was A and for 2 i thought it was C but both of those are wrong and i dont understand why.
i thought 1 was A because i found N to be 0 so f(c)=N so f(0)=0 seemed to sound correct to me so i thought it was always true
i thought 2 was C because yes 0 is between those numbers but other numbers could be between those two numbers too.
thank you
chad
Ok Ok... Forget all of the formal crap that your book is shoving down your throat right now. In your mind visualize a a graph of a function. Now, this function is a smooth curve that starts at -1 on the y axis to and goes to +1 on the y axis.
So, think about it...Does this curve - if it is unbroken - have to pass through 0 to go from -1 to 1?
Or better yet: On a sheet of graphing paper, put two points on the page. Put one at (-8,-1) and the other at (8,1). Now here's the chalenge: Without lifting your pencil from the paper, see if you can draw a curve from (-8,-1) to (8,1) and not cross the x axis. It's impossible! this is what the IVT says!
If the answer to the question "Does f pass through the x-axis (that is, does f equal zero) at some point on the interval?" is "yes", then the second consideration (which is the first exercise) is "Must the x-value for that pss-through point necessarily be zero itself?"
Think about that curve that you drew through the two points. Is it possible to make the curve go through (0,0)? If so, does it have to go through (0,0)? or can it just pass it by? If it can go through, how often will this occur for different pencilings (functions)? always? Sometimes? Never? Can you cross the x axis more than once on your way to the other point?
These type of questions are what the IVT are all about. If you still have trouble, let us know.