Question

Construct an example of a function f that is defined at every point in a closed interval, and whose values at the endpoints have opposite signs, but for which the equation f(x)=0 has no solution in the interval .

Any idea?

Answer 1

@popogirl95 I study calculus 1 and I was asking about intermediate-value theorem that says ((if f(x) is continuous on a closed interval [a,b] and C is any number between f(a) and f(b),inclusive, then there is at least one number x in the interval [a,b] such that f(x)=c))
an immediate consequence of this theorem is :
if f(x) is continuous on [a,b] and if f(a) and f(b) have opposite signs , then there is at least one solution of the equation f(x)=0 in the interval (a,b)

and the question is asking us to create an equation that fulfills ALL the conditions of this consequence however it does not have a solution in that interval IS THAT POSSIBLE ?!

Answer 2

how about some version of a unit step function?
http://www.efunda.com/math/unit_delta/unit_delta.cfm

try U(x)- 1/2 over the interval -1 ≤ x ≤ 1

Answer 3

Though it's the first time in my life i hear about such function i did my reading and my limited mind can understand that U(-1)-0.5=-0.5 and U(1)-0.5= 0.5 and there is no solution in between ...
Anyway it's far beyond my current understanding , so I'm not gonna bother myself alot about it ,,thanks again ...