Question

Does there exist a continuous function and differentiable function f(x) such that f(0)=−3 f(2)=−7 and f(x)=<−8 for all x in (02)? Answer(Yes or No)

Answer 1

No, not even a continuous function. Because for such a f you would have: \[(0 < x < 2 ⇒ f(x) \leq -8) ⇒ -3 = f(0) = \lim_{\substack{x→0\\x>0}}{f(x)} \leq -8\]