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For the function : ​ (a) Find the critical numbers of f (if any); (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. ​ Then use a graphing utility to confirm your results. ​

Identify the open intervals where the function is increasing or decreasing. ​

A ball bearing is placed on an inclined plane and begins to roll. The angle of elevation of the plane is radians. The distance (in meters) the ball bearing rolls in t seconds is . . Determine the speed of the ball bearing after t seconds. ​

The radius of a spherical balloon is measured to be 8 inches, with a possible error of 0.08 inch. Use differentials to approximate the maximum possible error in calculating the volume of the sphere. Round your answer to two decimal places. ​

Find all points of inflection of the graph of the function on the interval . Round your answer to three decimal places wherever applicable. ​

Determine whether Rolle’s Theorem can be applied to the function on the closed interval . If Rolle’s Theorem can be applied, find all values of c in the open interval such that . ​

Find the points of inflection and discuss the concavity of the function on the interval . ​

The radius of a spherical balloon is measured to be 8 inches, with a possible error of 0.08 inch. Use differentials to approximate the maximum possible error in calculating the volume of the sphere. Round your answer to two decimal places. ​

The graph of f is shown below. For which value of x is zero? ​ ​

Find a function f that has derivative  and with graph passing through the point . ​