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.