For the first one, remember the formula for the distance between two points? Try to write a formula for the distance between the origin and a point on your function, and then find the minimum of that function.
For the second question, you are on the right track. The intervals on which g'(x) is positive is where g is increasing, and where g'(x) is negative is where it is increasing. So you need to find out where g'(x) is 0.
To find out where g'(x)=0 there are several ways. There is a formula for quartics, but it is ugly. You could also graph the function and use a computer to find the zeros. One trick is to plug in r for x^2. This leaves you with a quadratic which is easy to factor. You can ignore the complex roots (if there are any).
Once you have the zeros, divide the real line at each zero check if g'(x) is positive or negative in every region.