My thoughts on the question.

The initial limit by direct substitution gets us 0/0, so we can use L Hospital rule to get f'(x) / 2 (x - x0). Note that f(x0) is a fixed number so differentiating this with respect to x gives a value of zero.

The denominator x - x0 will still equal zero if x = x0, so now we have to check the numerator f'(x) to see whether we need to do one more round of L Hospital rule.

If f'(x) is zero, then one more try of the L Hospital rule gives f''(x) / 2, and if the limit to x0 is 2 then f''(x0) / 2 = 2 so f''(x0) = 4 indicating a minimum point.

If f'(x) is positive or negative, then the limit tends to positive or negative infinity (since a non zero real number divided by 0 equals infinity) and the point is not even an extreme point. Since the limit is given at 2 and not +- infinity, this scenario is ruled out.

Therefore, the point has to be a minimum point.