Proving Unique Solution for f(x) = 0 on (a,b) with f'(x) ≠ 0

[msell2]

Assume that f is continuous on [a,b] and differentiable on (a,b). Assume also that f′(x) ≠ 0 on (a,b) and f(a) and f(b) have different signs. Show that the equation f (x) = 0 has a unique solution in (a, b).


I'm not really sure how to even start this proof. Do I need to use the Intermediate Value Theorem? Any help would be great!

 

[Ray Vickson]

Assume that f is continuous on [a,b] and differentiable on (a,b). Assume also that f′(x) ≠ 0 on (a,b) and f(a) and f(b) have different signs. Show that the equation f (x) = 0 has a unique solution in (a, b).


I'm not really sure how to even start this proof. Do I need to use the Intermediate Value Theorem? Any help would be great!

You actually need to prove two things: (1) there is a solution x in (a,b); and (2) there is only one such solution.

Do you know how to get (1)?

For (2): assume the contrary and see what happens.