A couple calc questions [optimization, max/min/inflection]

The depth of water at my favorite surfing spot varies from 4 to 1 feet, depending on time. Last sunday, high tide occurred at 5:00 a.m. and the next high tide occurred at 6:30 p.m. Obtain a cosine model describing the depth of the water as a function of time t in hours since 5 a.m. Sunday morning. Find the fastest rate at which the tide was rising on Sunday. Waht what time (after 5 a.m Sunday) did that first occur?

-I managed to get the cosine model, but I'm not sure where to go from there. I have y=1.5cos(13.5t)+5.5 as the model.

The function *f(x) = x *cos( ln* x* ) has infinitely many critical points. Find the smallest *x-*value **in the interval [1,∞)** for which *f(x)* a relative minimum point, a relative maximum point, and an inflection point.

-I'm not sure where to go if the function has infinitely many critical points. I know you're supposed to derivate, set the function equal to zero (finding critical points), and then check to see where it has max/mins, but if it has infinitely many I'm not sure where to go with that.

Find the dimensions of the right circular cylinder of greatest volume that can be inscribed in a sphere of radius *a*.

-I drew the diagram but I'm having a hard time trying to figure out where to go from there.