UPSC NDA Math Model Paper 8

Let, f(x) = sin(x + π /3) + cos(x + π /3)
Here, sin(x + π /3) + cos(x + π /3)
⇒√2[1∕√2(sin(x+π∕3) +1∕√2cos(x+π∕3)]
⇒√2[cos(π∕4)×sin(x+π∕3) +sin(π∕4)×cos(x+π∕3)]
⇒√2[sin(x+π∕3+π∕4)]
Now, we know sin x has maximum value at x = π/2
∴For maximum value of sin(x + π /3) + cos(x + π /3),
x + π/3 + π/4 = π/2
⇒ x = π/2 - 7π/12
⇒ x = π/12
Hence, option (4) is correct.