A differentiable function g is defined for all real numbers with g(1)=−2 and g(4)=5.
a. Explain why there must be at least one number c in the open interval (1,4) such that g(c)=0. Provide a clear logical argument that relies on the given information.
b. Determine whether there must also exist a number k in (1,4) such that g(k)=2. Justify your answer rigorously, citing any necessary properties of the function.