At what value of x is the function x2 + x + 1 minimum?

To find the value of `x` that minimizes the function `x^2 + x + 1`, we need to find the vertex of the parabola represented by this function. The vertex of a parabola of the form `ax^2 + bx + c` is given by `(-b/2a, f(-b/2a))`. In this case, `a = 1`, `b = 1`, and `c = 1`, so the vertex is located at `(-1/2, f(-1/2))`. To find `f(-1/2)`, we substitute `-1/2` for `x` in the function: `f(-1/2) = (-1/2)^2 + (-1/2) + 1 = 3/4` Therefore, the vertex of the parabola is `(-1/2, 3/4)`. Since the parabola opens upwards (the coefficient of `x^2` is positive), the value of `x` that minimizes the function is the x-coordinate of the vertex, which is `-1/2`. Therefore, the answer is `1`.