use the intermdeiate value theorem

Question

Use the Intermediate Value Theorem to prove that f(x) = x^ 2 − ((cos πx)/( x)) has at least one real zero in the interval [ 1/3 , 1].

Answer 1

Step 1 :

Intermediate value theorem :

If the function f  is continuous on the closed interval and let N  be any number between

and , where . Then there exists a number c  in such that .

Step 2 :

The function is and the point is .

The function is continuous for all values of x, except at

The function is continuous on the interval .

Substitute in .

Substitute in .

It follows that and .

Hence, the intermediate value theorem says there is a number c  such that .

The function has at least one root c  in the interval .

Solution :

The function has at least one root c  in the interval .