The absolute minimum value,of the function `f(x)=|x^(2)-x+1|+[x^(2)-x+1]`,where `[t]` denotes the greatest integer function,in the interval `[-1,2]`,is

Options

A. `(3)/(2)`
B. `(1)/(4)`
C. `(5)/(4),`
D. `(3)/(4)`

Answer

To find the absolute minimum value of the function \( f(x) = |x^2 - x + 1| + [x^2 - x + 1] \) in the interval \([-1, 2]\), we will follow these steps:

### Step 1: Analyze the function inside the absolute value
First, we need to analyze the quadratic function \( g(x) = x^2 - x + 1 \).

The vertex of the parabola represented by \( g(x) \) can be found using the formula \( x = -\frac{b}{2a} \):
- Here, \( a = 1 \) and \( b = -1 \).
- Thus, the vertex occurs at \( x = \frac{1}{2} \).

### Step 2: Evaluate \( g(x) \) at the endpoints and vertex
Next, we evaluate \( g(x) \) at the endpoints of the interval and at the vertex:
- \( g(-1) = (-1)^2 - (-1) + 1 = 1 + 1 + 1 = 3 \)
- \( g(2) = (2)^2 - (2) + 1 = 4 - 2 + 1 = 3 \)
- \( g\left(\frac{1}{2}\right) = \left(\frac{1}{2}\right)^2 - \left(\frac{1}{2}\right) + 1 = \frac{1}{4} - \frac{1}{2} + 1 = \frac{1}{4} - \frac{2}{4} + \frac{4}{4} = \frac{3}{4} \)

### Step 3: Determine the absolute values and greatest integer values
Now, we can find the values of \( f(x) \) at these points:
1. At \( x = -1 \):
\[ f(-1) = |g(-1)| + [g(-1)] = |3| + [3] = 3 + 3 = 6 \]
2. At \( x = 2 \):
\[ f(2) = |g(2)| + [g(2)] = |3| + [3] = 3 + 3 = 6 \]
3. At \( x = \frac{1}{2} \):
\[ f\left(\frac{1}{2}\right) = |g\left(\frac{1}{2}\right)| + [g\left(\frac{1}{2}\right)] = \left|\frac{3}{4}\right| + \left[\frac{3}{4}\right] = \frac{3}{4} + 0 = \frac{3}{4} \]

### Step 4: Compare the values
Now we compare the values of \( f(x) \) at the evaluated points:
- \( f(-1) = 6 \)
- \( f(2) = 6 \)
- \( f\left(\frac{1}{2}\right) = \frac{3}{4} \)

### Step 5: Conclusion
The absolute minimum value of \( f(x) \) in the interval \([-1, 2]\) is:
\[ \boxed{\frac{3}{4}} \]