Question

What is the max value of 4(sinx)^2+ 3(cosx)^2+ sin(x/2)+ cos(x/2)

Answer 1

sin(x) maxes out at 1, so does cos(x)

Answer 2

\[\Large \sin(x) \le 1\]
\[\Large \sin^2(x) \le 1^2\]
\[\Large \sin^2(x) \le 1\]
\[\Large 4*\sin^2(x) \le 4*1\]
\[\Large 4\sin^2(x) \le 4\]
hopefully this is making sense. If not, then let me know

Answer 3

this first bit simplifies straightaway ie by using Pythagoreas on \( 4\sin^2 x+ 3\cos^2 x\)

you can combine these bits: \(\sin{x\over2}+ \cos{x\over2}\) as
\(\sqrt{2} \{{ 1 \over \sqrt{2} } \sin{x\over2}+ { 1 \over \sqrt{2} } \cos{x\over2} \} = \sqrt{2} \sin ({x\over 2} + {\pi\over4})\)

then maybe some calculus...after you have tidied it up. looks messy.

personally, i'd stuff it in desmos.