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

Question

The number of solution of the equation

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

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Answer 1 · 2018-04-18T18:18:54

Zero

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.