Question

NOTE- If it is true, you need to prove it and If it is false, give a counterexample

f : [a, b] → R is continuous and in the open interval (a,b) differentiable.

a) If f(a) ≥ f(b), then exists a ξ ∈ (a,b) with f′(ξ) ≤ 0.(TRUE or FALSE?)

b) If f is reversable, then f −1 differentiable. (TRUE or FALSE?)
c) If f ′ is limited, then f is lipschitz. (TRUE or FALSE?)

Answer 1

a) The statement is true

If the function is a constant, it has zero slope everywhere and then we are done.
If it is not a constant it is decreasing somewhere at which the slope is negative. So in both of these cases the statement is true. Hence the statement is true.
b) The statement is true

Consider an invertible function f pocessing the derivative f^'. Since the function is invertible it is one-one and onto.
So we have,

Diffrentiating both sides with respect to x ,

By chain rule.
So,

So the condition is that

Any function violating this condition would raise a problem. But our function is one-one so it can not have zero slope anywhere. ( If it pocess zero slope somewhere there would be many points in [a,b] mapping to a single point in the range). So the derivative of the inverse function exists.

c) The statement is true

Let f^' be limited. i.e, let for some positive constant C.

let x and y belong to [a,b]. Now since f is diffrentiable, by mean value theorem , there exist a c in (x,y) so that

Now by taking modulus on both sides and by replacing by the heavier , we get the lipschitz condition,