Intermediate value theorem

[JP_Mtl]

Hey folks,

I have come across a puzzling problem that requires applying the Intermediate value theorem (IVT).

The question is as follows:

"Apply the intermediate value theorem to show that on the interval (0,3) there is a solution to the equation (x/2)=cos(x)."

I have only come across versions of the IVT that are applied to continuous functions... I tried graphing the equation provided using a graphing program and it seems to just be a vertical line slightly to the left of 1 on the x axis, probably corresponding to the value of x that is exactly equal to 2cosx (i.e. the equation slightly rearranged).

I am totally at a loss as to how to apply the IVT here - normally one would try to calculate endpoints (ie f(x) at x=0 and x=3 for instance) and show that for instance, because the function must be continuous, some value must be present between the endpoints, but in this case, that isn't possible since we're not even dealing with a function?!

 

[Steven G]

Hey folks,

I have come across a puzzling problem that requires applying the Intermediate value theorem (IVT).

The question is as follows:

"Apply the intermediate value theorem to show that on the interval (0,3) there is a solution to the equation (x/2)=cos(x)."

I have only come across versions of the IVT that are applied to continuous functions... I tried graphing the equation which equation? provided using a graphing program and it seems to just be a vertical line slightly to the left of 1 on the x axis, probably corresponding to the value of x that is exactly equal to 2cosx (i.e. the equation slightly rearranged).

I am totally at a loss as to how to apply the IVT here - normally one would try to calculate endpoints (ie f(x) at x=0 and x=3 for instance) and show that for instance, because the function must be continuous, some value must be present between the endpoints, but in this case, that isn't possible since we're not even dealing with a function?!

Please respond to question above in red.

if x=c is a solution that would mean that c/2 = cosc and hence their difference would be 0. Just define a function g(x)=(x/2)- cosx and show that there is a sol on (0,3) using the IVT.