Please help with example problems 1 and 2.

1.

Let f be continuous on [0,, b], differentiable on (a, b) and positive (i.e., > 0) for all :1: E ((1,5). Prove that there 3:53) = a; + Fla\" (Hint: consider the function F (:13) = (a: a) (3'1 5) f0\") and \"59 MVT for F (a?) to show the existence of such (2 E (a, b).) exists 0 E (a, b) such that 3} Use L'Hopital's Rule to evaluate and check your answers numerically: Sills: :12 (a) lilo ( ) x>O+ :E (b) 1' 1 i \"313% 811123? :82