Answer the following questions related to the relationship between the sign of the first derivative and the monotonic nature of a differentiable function.
a. (a) State the theorem that links the sign of f′(x) to the monotonicity of a differentiable function f(x) on an interval.
b. (b) Give a proof of the theorem stated in part (a) by using the Mean Value Theorem.
c. (c) Provide one explicit example of a differentiable function on an interval, verify the hypotheses of the theorem, and illustrate how the theorem correctly predicts the behaviour of the function on that interval.