Let f(x)=6sinxcosx+6cos2x−1f(x) = 6 \sin x \cos x + 6 \cos^2 x - 1f(x)=6sinxcosx+6cos2x−1.
Write f(x)f(x)f(x) in the form
asin2x+bcos2x+c a \sin 2x + b \cos 2x + c asin2x+bcos2x+cwhere a,b,a, b,a,b, and ccc are integers to be found.
Use the answer to part (a) to write f(x)f(x)f(x) in the form
Rsin(2x+α)+c R \sin (2x + \alpha) + c Rsin(2x+α)+cwhere R>0R > 0R>0 and 0<α<π20 < \alpha < \frac{\pi}{2}0<α<2π. Give the exact value of RRR and give the value of α\alphaα in radians to 3 significant figures.
Hence, or otherwise, (i) state the maximum value of f(x)f(x)f(x), (ii) find the second smallest positive value of xxx at which a maximum value of f(x)f(x)f(x) occurs. Give your answer to 3 significant figures.
