If the minimum value of f(x) = 5 x ^ 2 2 + α x ^ 5 , x > 0 , is 14...

From weighted $A . M . - G . M .$ inequality, we know

$\dfrac{5\left(\dfrac{{x}^{2}}{2}\right) + 2\left(\dfrac{\alpha}{2{x}^{5}}\right)}{5 + 2} \ge \left({\left(\dfrac{{x}^{2}}{2}\right)}^{5} \cdot {\left(\dfrac{\alpha}{2{x}^{5}}\right)}^{2}\right)\dfrac{1}{7}$

$⇒ \dfrac{5{x}^{2}}{2} + \dfrac{\alpha}{{x}^{5}} \ge \dfrac{7}{2}\left(\alpha\right)\dfrac{2}{7}$

Given that the least value of $\dfrac{5{x}^{2}}{2} + \dfrac{\alpha}{{x}^{5}}\text{is}14$

i.e. $\dfrac{7}{2}\left(\alpha\right)\dfrac{2}{7} = 14 ⇒ \left(\alpha\right)\dfrac{1}{7} = 2 ⇒ \alpha = 128$