Question 6 - A-Level Maths

Exam Board	Edexcel
Module	AEA (Advanced Extension Award)
Year	2007
Session	June
Topic	Differentiation Applications
Type	Optimization with constraints

Find an expression, in terms of \(x\), for the area \(A\) of \(R\).
Show that \(\frac { \mathrm { d } A } { \mathrm {~d} x } = \frac { 1 } { 4 } ( \pi - 2 x - 2 \sin x ) \sec ^ { 2 } \frac { x } { 2 }\).
Prove that the maximum value of \(A\) occurs when \(\frac { \pi } { 4 } < x < \frac { \pi } { 3 }\).
Prove that \(\tan \frac { \pi } { 8 } = \sqrt { } 2 - 1\).
Show that the maximum value of \(A > \frac { \pi } { 4 } ( \sqrt { } 2 - 1 )\).