1. I'm not sure, but I'm not familiar with this wording... does extreme values include inflexion points too?

Because the graph of f(x) = tan(x) doesn't have any maxima or minima...

or I'm overlooking something.

2. Make a sketch.

Let's say the length of the rectangle is l and the width w.

Area of rectangle =

Draw a line from the centre of the semicircle to the vertex of the rectangle. This is the radius, right?

We get:

Use that in the above equation, and we get:

Now, find the derivative of Area with respect to w, and set this to 0 and solve for w to get the width when the area is a maximum.