Question #04706

1 Answer
Oct 5, 2016

I assume that you have Rolle's Theorem and the Mean Value Theorem to work with.

Explanation:

#f# is continuous and differentiable on its domain.

If there were #x_1 != x_2# with #f(x_1)=f(x_2)#, then there would be a #c# in #(x_1, x_2)# with #f'(c) = 0# (Rolle's or MVT)

But, #f'(x)# is (strictly) negative on the domain of #f#.

Therefore, there cannot be #x_1 != x_2# with #f(x_1)=f(x_2)#.

So #f# is one-to-one and has an inverse.