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Maths-

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Question

$text If end text 0 less than alpha less than fraction numerator pi over denominator 2 end fraction text and end text f left parenthesis alpha right parenthesis equals pi to the power of sec to the power of 2 end exponent invisible function application alpha end exponent c o s to the power of 2 end exponent invisible function application alpha plus pi to the power of cosec to the power of 3 end exponent invisible function application alpha end exponent s i n to the power of 2 end exponent invisible function application alpha comma text end text$ then

$f (α) < π^{2}$
$f (α) > π^{2}$
$f (α) < π$
$f (α) = π^{2}$

The correct answer is: $f left parenthesis alpha right parenthesis greater than pi to the power of 2 end exponent$

AM ≥ GM. Also
$f open parentheses x close parentheses minus 2 open parentheses A M A f subscript m end subscript sec to the power of 2 end exponent invisible function application alpha cos to the power of 2 end exponent invisible function application alpha text and end text minus cosec to the power of 2 end exponent invisible function application alpha sin to the power of 2 end exponent invisible function application alpha close parentheses$
$text Using end text A M greater than G M text we see that end text$
$f left parenthesis x right parenthesis greater or equal than 2 open parentheses G M text of end text pi to the power of sec to the power of 2 end exponent invisible function application alpha end exponent c o s to the power of 2 end exponent invisible function application alpha text and end text pi to the power of cosec to the power of 2 end exponent invisible function application alpha end exponent s i n to the power of 2 end exponent invisible function application alpha close parentheses$
$equals 2 square root of open parentheses pi to the power of sec to the power of 2 end exponent invisible function application alpha end exponent c o s to the power of 2 end exponent invisible function application alpha close parentheses open parentheses pi to the power of cosec to the power of 2 end exponent invisible function application alpha end exponent s i n to the power of 2 end exponent invisible function application alpha close parentheses end root equals s i n invisible function application 2 alpha open parentheses pi to the power of 2 divided by s i n to the power of 2 end exponent invisible function application left parenthesis 2 alpha right parenthesis end exponent close parentheses$
$text Now let end text g left parenthesis x right parenthesis equals x pi to the power of 2 divided by x to the power of 2 end exponent end exponent text is a decreasing function for end text x element of left parenthesis 01 right parenthesis text . end text$
$text Also for end text 0 less than alpha less than pi divided by 2 comma s i n invisible function application 2 alpha element of left parenthesis 01 right parenthesis$
$text Therefore, end text g left parenthesis s i n invisible function application 2 alpha right parenthesis greater than g left parenthesis 1 right parenthesis greater than pi to the power of 2 end exponent$

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