The maximum value of 4sin2⁡x+3cos2⁡x+sin⁡x2+cos⁡x2 is

Maximum value of 4sin2⁡x+3cos2⁡x i.e., sin2⁡x+3 is 4and that of sin x2+cos⁡x2 is 12+12=2,both attained  at x=π2. Hence, the given function has maximum value 4+2