Question

Find range of

f(x)=3cosx+4sinx+10

Answer 1

Let y = 3 cosx + 4 sinx +10
⇒ (y-10) = 3 cosx + 4 sinx
Recall that for all real values of x,
−√(a^2+b^2) ≤ ( a sin x + b cos x ) ≤ √(a^2+b^2) → (1)
⇒ −√(3^2 + 4^2) ≤ (y-10) ≤ +√(352 + 4^2) [From (1)]
⇒ −5 ≤ ( y -10 ) ≤ 5
⇒ −5 +10 ≤ [( y -10 ) +10] ≤ 5 +10
⇒ 5 ≤ y ≤15
⇒ 5≤ (4 sin x + 3 cos x +10) ≤ 15
Hence the range is [5, 15].