Which of the following must be true?

Question

For the differentiable function on the interval [1, 3]

f(1)=-2   f(2)=4  f(3)=-2

Which of the following must be true?

I. There must be at least one point on the interval where is zero.
II. There must be at least two points on the interval where is zero.
III. There must be at least two points on the interval where is zero.

Answer 1

f(1)=-2 is he same as saying that this function goes through (1,-2). Likewise, the function must also go through (2,4) and (3,-2). 

 

Since he function is differentiable on this entire interval, the graph must also be continuous on the interval.

 

So if yo would plot these 3 points and try to draw a continuous graph connecting these points, you would have to go through the x axis somewhere between x=1 and x=2 and then again from x=2 to x=3.

 

This means there is a value between 1 and 2 that has a y coordinate of zero (x intercept) and another zero between 2 and 3.

 

Therefore, there must be at least 2 points on the interval 1 to 3 where there is a zero.

 

Hope this helped.