y=√5[(2√5/5)sinx+(√5/5)cosx]

Note: This √5 is the square root of the sum of the squares of the antecedents of sinx and cosx. For example, y=asinx+bcosx should be √ (A 2+B 2).

The reason for this is to construct a trigonometric function to facilitate the following calculation.

Let cosβ=2√5/5, sinβ=√5/5 tanβ= 1/2.

y=√5[cosβsinx+sinβcosx]

=√5sin(β+x) tanβ= 1/2

1≥sin(β+x)≥- 1

get

√5≥y≥-√5

I hope you are satisfied.

In fact, we can find that the maximum value is √ (A 2+B 2) and the minimum value is -√ (A 2+B 2).

thank you