Does there exist a continuous function f(x) such that f(0)=9, f(2)=3 and f′(x)≤−9 for all x in (0,2)?

Answer 1

No - we can justify this using the Mean Value Theorem.

The mean value theorem states that for any differentiable function on the interval (a,b), there must be some point, let's call it "c", such that c = (f(b) - f(a))/(b-a) ;or, essentially the slope of a line connecting the two points.

The slope from (0,9) to (2,3) is -3.

Since this function is continuous and is implied to be differentiable on the interval, the condition of f'(x)<=9 simply cannot occur, since the slope between the two points, -3, is not less than -9.