Solve Optimisation Problem: 2 Attached Files & Markscheme Explained

[nokia8650]

http://img119.imageshack.us/img119/3906/optdp4.th.jpg

http://img246.imageshack.us/img246/7067/opt1rf0.th.jpg See two attached files for the problem and markscheme

My question is, why must one differentiate in part d; the markscheme finds dS/da and then says the function is increasing, before finding the boundary values. What is the need for this?

Thanks

 

[Defennder]

That's because for any given continuous function, the maximum of that function in any given interval may or may not be at the endpoints. S(alpha) in this question might not attain its maximum value at either endpoints, but at some point x1 in the interval pi/4 < alpha < 1. It is only after differentiating and showing that S is strictly increasing in that interval then it is possible to conclude that the max value of S is at the rightmost endpoint.

If, instead S is strictly decreasing, then the max value of S would be attained at pi/4, the left endpoint. If it is neither strictly increasing or decreasing throughout that interval, then it would not be possible to solve the problem in that manner.

 

[nokia8650]

Thanks for the help. They have given the value of S in a given range - would it therefore not be better to give the value of S at the right endpoint, since S is increasing, meaning that at the right endpoint, the value will be greater than at the left?

Thanks

 

[Defennder]

Yes, that is so.