- An extrema (minimum/ maximum) is where the first derivative of the function is equal to zero:
$f'(x)=0$ $2x-\frac{a}{x^2}=0$
We have $x=2$, so:$2(2^3)-a=0$
Therefore, $a$ would be 16.
As pointed out in comments, you should check whether the answer is correct:
$min\{f(x)=x^2+\frac{16}{x}\}=12$ and local minimum happens at $x=2$.
Looking at the plot might make it clearer: 
- An inflection point happens where the second derivative is equal to zero as well as all other higher order derivatives , meaning if the second derivative is zero but the fourth derivative is non-zero, for example, then the point is not an inflection point. What we have:
$f''(x)=2+\frac{2a}{x^3}=0$ Thus, at $x=1$, $a$ would be -1.
