Question

determine wether we can use the intermediate value therom to justify that there is at least one root of the equation sin(Pi x) - 1/x in each intervals: a) [1,3] b) [1.5,2.5] c)[-1,1]

Answer 1

you can use the thm if the signs change for each endpoint
--> f(a) is positive and f(b) is negative
or other way around

Answer 2

so your saying for the therom to work of the points provided one has to be positive and the other nagative?

Answer 3

AND that the function is continuous on the entire interval.

Answer 4

For example, the function f(x) = 1/x is positive for x = 1 and negative for x = -1. But there is no value \( a \in (-1,1) \) such that \( f(a) = 0 \).