The minimum value of $a$ secx $+ b$ cosecx is

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(a^{2 / 3} - b^{2 / 3})^{2 / 3}

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(a^{2 / 3} + b^{2 / 3})^{3 / 2}

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a^{2} b^{2}

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Solution

The correct option is B $(a^{2 / 3} + b^{2 / 3})^{3 / 2}$
$y = a s e c x + b c o s e c x$
$y^{'} = a s e c x t a n x - b c o s e x x c o t x$
$b c o s e c x c o t x = a s e c x t a n x$
$b c o s^{3} x = a s i n^{3} x$
$(\frac{b}{a})^{\frac{1}{3}} = t a n x$
$= a (1 + (\frac{b}{a})^{\frac{2}{3}})^{\frac{1}{2}} + b (1 + (\frac{b}{a})^{\frac{2}{3}})^{\frac{1}{2}} (\begin{matrix} s e c x = & 1 + t a n^{2} x c o s e c x = & 1 + c o t^{2} x \end{matrix})$
$= a^{\frac{2}{3}} (a^{\frac{2}{3}} + b^{\frac{2}{3}})^{\frac{1}{2}} + b^{\frac{2}{3}} (a^{\frac{2}{3}} + b^{\frac{2}{3}})^{\frac{1}{2}}$
$= (b^{\frac{2}{3}} + a^{\frac{2}{3}})^{\frac{3}{2}}$

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