InterviewSolution
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InterviewSolution
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The least value of `2^(sinx)+2^(cosx)`, isA. `2^(1//sqrt(2))`B. `2^(1-2^(-1//2))`C. `2^(1+2^(-1//2))`D. `2^(1-sqrt(2))` |
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Answer» Correct Answer - B Using `A.M.geG.M.,` we have `(2^(sinx)+2^(cosx))/(2)ge(2^(sinx)xx2^(cosx))^(1//2)` `implies" "2^(sinx)+2^(cosx)ge2{2^((1)/(2)(sinx+cosx))}` `implies" "2^(sinx)+2^(cosx)ge2^(1+(1)/(2)(sinx+cosx))" "...(i)` Now, `-sqrt(2)lesinx+cosxlesqrt(2)" for all "x` `implies" "-(1)/(sqrt(2))le(1)/(2)(sinx+cosx)le(1)/(sqrt(2))` `implies" "1-(1)/(sqrt(2))le1+(1)/(2)(sinx+cosx)le1+(1)/(sqrt(2))` `implies" "2^(1-(1)/(sqrt(2)))le2^(1+(1)/(2)(sinx+cosx))le2^(1+(1)/(sqrt(2)))" "...(ii)` From (i) and (ii), we get `2^(sinx)+2^(cosx)ge2^(1-(1)/(sqrt(2)))" for all "x in R.` Hence, the least value of `2^(sinx)+2^(cosx)" is "2^(1-(1)/(sqrt(2)))` |
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