To find the minimum value of the parabola given by the equation
\[ y = x^2 + x + \frac{11}{4}, \]
we can use the formula for the vertex of a parabola, which occurs at \( x = -\frac{b}{2a} \), where \( a \) is the coefficient of \( x^2 \), and \( b \) is the coefficient of \( x \).
Here, \( a = 1 \) and \( b = 1 \).
Calculating the x-coordinate of the vertex:
\[ x = -\frac{b}{2a} = -\frac{1}{2 \times 1} = -\frac{1}{2}. \]
Next, we substitute \( x = -\frac{1}{2} \) back into the equation to find the minimum value of \( y \):
\[ y = \left(-\frac{1}{2}\right)^2 + \left(-\frac{1}{2}\right) + \frac{11}{4}. \]
Calculating each term:
\[ \left(-\frac{1}{2}\right)^2 = \frac{1}{4}, \] \[ -\frac{1}{2} = -\frac{2}{4}, \] \[ \frac{11}{4} = \frac{11}{4}. \]
Now, combine these values:
\[ y = \frac{1}{4} - \frac{2}{4} + \frac{11}{4} = \frac{1 - 2 + 11}{4} = \frac{10}{4} = \frac{5}{2}. \]
Thus, the minimum value of the given parabola is
\[ \boxed{\frac{5}{2}}. \]
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