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) f rises strictly monotonously ⇐ ∀x ∈ (a, b) : f ′(x) > 0. (TRUE or FALSE?)
b) f is constant ⇐⇒ ∀x∈(a,b): f′(x)=0 (TRUE or FALSE?)
c) If f is reversable, f has no critical point. (TRUE or FALSE?)
d) If a is a “minimizer” of f, then f ′(a) = 0. (TRUE or FALSE?)
e) If f(a) ≥ f(b), then exists a ξ ∈ (a,b) with f′(ξ) ≤ 0. (TRUE or FALSE?)

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

Answer 1