f(x) = x^{2} + x + 1

Hello!

Make a perfect square:f(x) = x^2 + x + 1 = (x + 1/2)^2 - 1/4 + 1 = (x + 1/2)^2 +3/4.

(x + 1/2)^2 >= 0, and =0 only for x=-1/2.

So f has a minimum at x=-1/2 and the minimum value is f(-1/2) = 3/4.

Answer: minimum value is 3/4 (reached at x=-1/2).

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