Question

Discuss the extreme values of the function f(x) = a sin x + b cos x.

Answer 1

This proof can be done differently if you can understand vectors

Lets say U, V are two vectors such that  

U= sinx (i) + cosx (j)  

V= a (i)  + b (j)

magnitudes

|U|  = 1

|V|  = (a^2 + b^2)^(0.5)

f(x) = U.V = a Sinx + b Cosx  = |U||V| Cos t

Where t is angle between the vectors U and V

Maximum and minimum of U.V are +|U||V| and -|U||V| respectively

as maximum and minimum values of Cost are +1, -1 respectively.