The expression `2^(sintheta)+2^(-costheta)` is minimum when `theta` is equal to

Options

A. `2npi+(pi)/(4)`
B. `2npi+(7pi)/(4)`
C. `n pipm(pi)/(4)`
D. none of these

Answer

B

Explanation

We have,
`2^(sintheta)+2^(-costheta)ge2sqrt(2^(sintheta-costheta))" "["Using"AMgeGM]`
`implies2^(sintheta)+2^(-costheta)ge2^(1+(1)/(sqrt2)sin(0,(pi)/(4)))`
Thus, `2^(sintheta)+2^(-costheta)` is minimum when `sin(theta-pi//4)` is minimum i.e.
`sin(theta-(pi)/(4))=-1`
`impliestheta-(pi)/(4)=2npi+(3pi)/(2),ninZ.`
`impliestheta=2npi+(7n)/(4),n inZ.`