The number of solution of the equation 3^(sin2x+2cos2x)+ 3^(1-sin2x+2sin2x)=28 in [0,2π]?

The number of solution of the equation

#3^(sin2x+2cos2x)+ 3^(1-sin2x+2sin2x)=28# For #x in [0,2π]#

1 Answer
Apr 18, 2018

Zero

Explanation:

The maximum value of
#sin2x+2cos2x = sqrt5 sin(2x+tan^-1 2)#
is #sqrt5#, So, the maximum possible value of

#3^(sinx+2cos2x)#

is #3^sqrt5~~11.7#

On the other hand, the maximum possible value of #1-sin2x+2sin2x = 1+sin2x# is 2, so that the maximum possible value of this term is #9#

This means that the sum can not exceed #~~20.7#, let alone equal 28.