Key Concepts and Formulas

• Quadratic Functions: The vertex of a parabola $ax^2 + bx + c$ occurs at $x = -\frac{b}{2a}$. The minimum (if $a>0$) or maximum (if $a<0$) value is found by substituting this $x$ into the function.
• Absolute Value Function: $|t| = t$ if $t \ge 0$, and $|t| = -t$ if $t < 0$.
• Greatest Integer Function: $[t]$ is the greatest integer less than or equal to $t$. For any real number $t$, $t-1 < [t] \le t$.
• Function Composition: Analyzing the inner function's range is crucial before evaluating the outer function.

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Step-by-Step Solution:

Step 1: Analyze the Inner Function $g(x)$ and its Range Let $g(x) = x^2 - x + 1$. We need to find the range of $g(x)$ for $x \in [-1, 2]$. This is a quadratic function, and its graph is a parabola opening upwards (since the coefficient of $x^2$ is positive). The minimum value occurs at the vertex. We can find the vertex by completing the square: $g(x) = x^2 - x + 1 = \left(x^2 - x + \frac{1}{4}\right) - \frac{1}{4} + 1 = \left(x - \frac{1}{2}\right)^2 + \frac{3}{4}$ The vertex is at $x = \frac{1}{2}$. Since $x = \frac{1}{2}$ lies within the interval $[-1, 2]$, the minimum value of $g(x)$ in this interval is $g\left(\frac{1}{2}\right) = \frac{3}{4}$.

To find the maximum value, we evaluate $g(x)$ at the endpoints of the interval: $g(-1) = (-1)^2 - (-1) + 1 = 1 + 1 + 1 = 3$. $g(2) = (2)^2 - (2) + 1 = 4 - 2 + 1 = 3$. Thus, the range of $g(x)$ for $x \in [-1, 2]$ is $\left[\frac{3}{4}, 3\right]$.

Explanation: By analyzing the quadratic $g(x)$, we determine its minimum and maximum values within the given domain. This range is essential for simplifying the subsequent parts of the function.

Step 2: Simplify the Function $f(x)$ The given function is $f(x) = |x^2 - x + 1| + [x^2 - x + 1]$, which can be written as $f(x) = |g(x)| + [g(x)]$. From Step 1, we know that $g(x) \in \left[\frac{3}{4}, 3\right]$ for $x \in [-1, 2]$. Since all values in this range are positive, $|g(x)| = g(x)$. Therefore, $f(x) = g(x) + [g(x)]$.

Explanation: Since the expression inside the absolute value is always positive in the given interval, the absolute value function can be removed, significantly simplifying the problem.

Step 3: Find the Minimum of $h(y) = y + [y]$ for $y \in \left[\frac{3}{4}, 3\right]$ Let $y = g(x)$. We need to find the minimum value of $h(y) = y + [y]$ where $y \in \left[\frac{3}{4}, 3\right]$. We consider sub-intervals based on the value of $[y]$:

• Case 1: $y \in \left[\frac{3}{4}, 1\right)$ In this interval, $[y] = 0$. So, $h(y) = y + 0 = y$. The minimum value in this sub-interval occurs at the smallest value of $y$, which is $y = \frac{3}{4}$. The minimum value is $h\left(\frac{3}{4}\right) = \frac{3}{4}$. This occurs when $g(x) = \frac{3}{4}$, which is at $x = \frac{1}{2}$.

• Case 2: $y \in [1, 2)$ In this interval, $[y] = 1$. So, $h(y) = y + 1$. Since $h(y)$ is an increasing function of $y$, its minimum value in this interval occurs at $y = 1$. The minimum value is $h(1) = 1 + 1 = 2$.

• Case 3: $y \in [2, 3)$ In this interval, $[y] = 2$. So, $h(y) = y + 2$. Again, $h(y)$ is an increasing function of $y$. Its minimum value in this interval occurs at $y = 2$. The minimum value is $h(2) = 2 + 2 = 4$.

• Case 4: $y = 3$ In this case, $[y] = 3$. So, $h(3) = 3 + 3 = 6$. This occurs when $g(x) = 3$, which is at $x = -1$ or $x = 2$.

Explanation: The greatest integer function creates steps in the function's graph. By examining intervals where $[y]$ is constant, we can analyze $h(y)$ as a simpler (linear) function and find its minimum in each interval.

Step 4: Determine the Absolute Minimum Value Comparing the minimum values obtained from each case: $\frac{3}{4}$, $2$, $4$, and $6$. The smallest of these values is $\frac{3}{4}$. This minimum occurs when $y = g(x) = \frac{3}{4}$, which corresponds to $x = \frac{1}{2}$. Since $x = \frac{1}{2}$ is within the given interval $[-1, 2]$, this is the absolute minimum value of $f(x)$.

Explanation: We compare the minimum values found in each segment of the domain of $y$ to identify the overall absolute minimum.

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Common Mistakes & Tips:

• Range of $g(x)$ is key: Always determine the precise range of the inner function $g(x)$ over the given interval before simplifying $f(x)$.
• Interval for $x$ vs. $y$: Be mindful of the domain of $x$ and the corresponding range of $y=g(x)$. Ensure that the $x$-values yielding the minimum are within the original interval.
• Greatest Integer function behavior: Remember that $[y]$ is constant over intervals like $[n, n+1)$ but changes at integer values. The function $y + [y]$ is generally increasing but has jumps.

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Summary:

We began by analyzing the quadratic function $g(x) = x^2 - x + 1$ over the interval $[-1, 2]$, determining its range to be $\left[\frac{3}{4}, 3\right]$. Because $g(x)$ is always positive in this range, the absolute value $|g(x)|$ simplified to $g(x)$. The problem then reduced to finding the minimum of $h(y) = y + [y]$ for $y \in \left[\frac{3}{4}, 3\right]$. By considering sub-intervals based on the integer values of $y$, we found that the minimum value of $h(y)$ is $\frac{3}{4}$, occurring when $y = \frac{3}{4}$ (which corresponds to $x = \frac{1}{2}$).

The final answer is $\boxed{\text{(A) } \frac{3}{4}}$.