MLE of independent sample for the PDF $f(x;\theta) =e^{-(x-1)^2/(2\theta x)}/\sqrt{2\theta\pi x^3}$

Question

If the pdf is given as $$f(x;\theta) = \frac{1}{\sqrt{2\theta\pi x^3}}e^{-(x-1)^2/(2\theta x)}$$

I calcualted the MLE of the above given pdf and it's coming out to be $$ \hat \theta = \frac1n\sum X_i - 2 + \frac1n\sum\frac{1}{X_i}$$

How do I verify if the this estimate is correct?

I have posted my solution here: https://i.stack.imgur.com/BdSm4.jpg