A differentiable function is defined for all real numbers with and .

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

A differentiable function

g

is defined for all real numbers with

g(1)=-2

and

g(4)=5

. (1) 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. (2) Determine whether there must also exist a number

k

(1,4)

such that

g(k)=2

. Justify your answer rigorously, citing any necessary properties of the function.