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Maths-

General

Easy

Question

The minimum value $m space equals bold space 2 cos space theta plus fraction numerator 1 over denominator sin invisible function application theta end fraction plus square root of 2 tan space theta space text in end text open parentheses 0 comma pi over 2 close parentheses$ is

$2 + \sqrt{2}$
$3 \sqrt{2}$
$2 \sqrt{3}$
$3 + \sqrt{2}$

hint Hint:

In this question, we have to find the minimum value of $m space equals bold space 2 cos space theta plus fraction numerator 1 over denominator sin invisible function application theta end fraction plus square root of 2 tan space theta space text in end text open parentheses 0 comma pi over 2 close parentheses$ . We will be using the property AM>GM and further simplify it to find the required value.

The correct answer is: $3 square root of 2$

$m equals 2 space cos space ϑ plus fraction numerator 1 over denominator sin space ϑ end fraction plus square root of 2 tan space ϑ sin c e comma space ϑ element of open parentheses 0 comma pi over 2 close parentheses S o comma space sin space ϑ greater than 0 comma space cos space ϑ greater than 0 comma tan space ϑ greater than 0 u sin g space A M thin space greater or equal than G M space p r o p e r t y. rightwards double arrow fraction numerator 2 space cos space ϑ plus begin display style fraction numerator 1 over denominator sin space ϑ end fraction end style plus square root of 2 tan space ϑ over denominator 3 end fraction greater or equal than 3 open parentheses 2 square root of 2 close parentheses to the power of bevelled 1 third end exponent rightwards double arrow 2 space cos space ϑ plus fraction numerator 1 over denominator sin space ϑ end fraction plus square root of 2 tan space ϑ greater or equal than 3 open parentheses square root of 2 cubed close parentheses to the power of bevelled 1 third end exponent rightwards double arrow 2 space cos space ϑ plus fraction numerator 1 over denominator sin space ϑ end fraction plus square root of 2 tan space ϑ greater or equal than 3 square root of 2 S o comma space m i n i m u m space v a l u e space i s space 3 square root of 2.$

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