QJEEM22Ma7f83: JEE Main 2022 Mathematics - Class 12 JEEM22Ma7f83 Solution: If the minimum value of , is 14 , then the value of is equal to :

f(x) = {{5{x^2}} \over 2} + {\alpha \over {{x^5}}}\,\,\{ x > 0\}

$f'(x) = 5x - {{5\alpha } \over {{x^6}}} = 0$

$\Rightarrow x = {(\alpha )^{{1 \over 7}}}$

$f{(x)_{\min }} = {{5{{(\alpha )}^{{2 \over 7}}}} \over 2} + {\alpha \over {{\alpha ^{{5 \over 7}}}}} = 14$

${5 \over 2}{\alpha ^{{2 \over 7}}} + {\alpha ^{{2 \over 7}}} = 14$

${7 \over 2}{\alpha ^{{2 \over 7}}} = 14$

$\alpha = 128$