Sorry, wasn't sure how to describe the problem in the title. The problem statement, all variables and given/known data Okay, this problem has really been bugging me for a while. It's a question that I thought of when I was daydreaming in class, but now I can't stop thinking about it. I've been going crazy for days Here is the problem: - Consider the following information to determine the answer to the question afterward. f (0) and g (0) = 0 f '(0) and g '(0) = 1 f (h) and g (h) = 1 f '(h) and g '(h) = 0 h is positive constant greater than 1 f (x) and g (x) are always concave down on the interval (0, h) f (x) and g (x) are both continuous on the interval (0, h) f (x) and g (x) are both always differentiable on the interval (0, h) On the interval (0,h), do the graphs of f (x) and g (x) have to be identical? If not, give the algebraic expression of two functions that satisfy the above conditions and also have nonidentical graphs on the interval (0,h). The attempt at a solution I don't believe two functions necessarily have to be identical on the interval with those conditions. However, I can't find two functions to give an example My first attempt was to use f(x) = sinx as one of the functions, since it satisfies these conditions when h=pi/4. when f(x) = sinx, and h=pi/4 f (0) = 0 f '(0) = 1 f (pi/4) = 1 f '(pi/4) = 0 Now I just needed to find a function g(x) with the same conditions, but has a curve nonidentical to sinx on the interval (0, pi/4). I figured if I manipulate the semi-circle equation, I can obtain a graph with the above characteristics. So I set g(x) = a√(b-cx^2) + d, which is a transformation of the semi-circle equation. However, after hours upon hours of attempts, I cannot find the values of a, b, c & d to satisfy all three conditions! For example, I can easily get g(0) = 0, g'(0) = 1, and g (pi/4) = 1, but then g '(pi/4) would equal something other than 0. What is happening? I also tried other functions like the cubic function, function raised to the 1/3 power, etc. but nothing works. There's always one condition unfulfilled. - Am I wrong in my thinking that two functions with the given conditions are not necessarily identical? I asked my GSI and he gave me some philosophical nonsense and told me I can figure it out on my own (he's obviously wrong). PS: Could you explain it in a way that a first-semester Calculus student can understand?