A student is reviewing the Extreme Value Theorem (EVT) and writes the following four statements about a real-valued function f.
I. If f is continuous on a closed interval [a,b], then f attains exactly one absolute maximum and exactly one absolute minimum on [a,b].
II. If f is continuous on the bounded interval (a,b), then f attains both an absolute maximum and an absolute minimum on (a,b).
III. If f is continuous on [a,b] and differentiable on (a,b), then at least one absolute extremum of f occurs at a critical point in (a,b).
IV. If f has a critical point at x=c, then f must have either a local maximum or a local minimum at x=c.
For each statement answer the questions that follow.
a. Indicate whether Statement I is true or false and provide a complete justification. If the statement is false, supply a counterexample and explain why it fails.
b. Indicate whether Statement II is true or false and provide a complete justification. If the statement is false, supply a counterexample and explain why it fails.
c. Indicate whether Statement III is true or false and provide a complete justification. If the statement is false, supply a counterexample and explain why it fails.
d. Indicate whether Statement IV is true or false and provide a complete justification. If the statement is false, supply a counterexample and explain why it fails.