Math

does there exist a number in the interval such that is the slope of the secant line passing through the points and ? Is your answer consistent with the statement of the Mean Value Theorem?

 (2--6) Determine the intervals on which the function is increasing and intervals on which the function is decreasing. Check your answers by graphing (grapher) the corresponding functions. Include copies of these graphs.

.

 restricted to .

 restricted to .

 restricted to .

 restricted to positive .

 (7--10) Determine the intervals on which the function is concave up and intervals on which the function is concave down. Before you submit your solutions, check your answers by graphing the corresponding functions. No need to include these graphs.

.

 .

 defined on the interval .

 .

 Does the function , defined for , satisfy the hypotheses of the Mean Value Theorem? Find such that is the slope of the secant line passing through , and .

 Discuss but do not sumbit. Using the secant definition of concavity, explain that if the function

defined for is increasing for all , then is concave up.

 Discuss but do not submit. Assume that on . Show that