Finding parameter values Problem:

What value of $a$ makes $$f(x)=x^2 + \frac {a}{x}$$ have (a) a minimum at $x=2$, (b) an inflection point at $x=1$?
(a) and (b) are separate and not dependent upon each other.

What I have done:
Not much. I wrote the derivative to try and find where $x=0$, but that is where I got stuck.
$$0=2x-ax^{-2} .$$
I'm not really sure what to do.
(a) Take first derivative of $f(x)$, set it equal to 0, replace $x$ by 2 , and solve for $a$.

(b) Take second derivative of $f(x)$, set it equal to 0, replace $x$ by 1, and solve for $a$.
$F(x)=x^2+a/x$

$F'(x)=2x-a/x^2$

let $f'(x)=0$ and put $x=2$

$a=8$

$F"(x)=2+2a/x^3$ and let $F"(x)=0$ and put $x=1$

$a =-2$