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The given function is \( f(x) = 2x^2 - 6x + 5 \), and the question asks for the minimum value of the function.

### Steps to find the minimum value of a quadratic function:

1. **Identify the quadratic form:** The given equation \( f(x) = 2x^2 - 6x + 5 \) is in the standard form \( ax^2 + bx + c \), where:
   - \( a = 2 \)
   - \( b = -6 \)
   - \( c = 5 \)

2. **Find the vertex (minimum point for a parabola when \( a > 0 \)):**
   The x-coordinate of the vertex of a parabola is given by the formula:
   \[
   x = \frac{-b}{2a}
   \]
   Substituting \( a = 2 \) and \( b = -6 \):
   \[
   x = \frac{-(-6)}{2(2)} = \frac{6}{4} = \frac{3}{2}
   \]

3. **Find the minimum value (y-coordinate at the vertex):**
   To find the minimum value, substitute \( x = \frac{3}{2} \) into the function \( f(x) \):
   \[
   f\left(\frac{3}{2}\right) = 2\left(\frac{3}{2}\right)^2 - 6\left(\frac{3}{2}\right) + 5
   \]
   Simplifying each term:
   \[
   = 2\left(\frac{9}{4}\right) - 6\left(\frac{3}{2}\right) + 5
   \]
   \[
   = \frac{18}{4} - 9 + 5 = \frac{9}{2} - 9 + 5
   \]
   \[
   = \frac{9}{2} - \frac{18}{2} + \frac{10}{2} = \frac{9 - 18 + 10}{2} = \frac{1}{2}
   \]

Thus, the minimum value of \( f(x) \) is \( \frac{1}{2} \).

### Correct Answer:
The correct option is **B**: \( f(x) = \frac{1}{2} \).

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Would you like a more detailed explanation or have any further questions?

### Related Questions:
1. How do you find the maximum value of a quadratic function when \( a < 0 \)?
2. What is the significance of the vertex in the graph of a quadratic function?
3. How do you convert a quadratic function to vertex form?
4. What role does the discriminant play in quadratic equations?
5. How does the value of \( a \) affect the shape of a parabola?

### Tip:
When dealing with quadratic functions, remember that if the coefficient of \( x^2 \) (i.e., \( a \)) is positive, the parabola opens upwards, and the vertex gives the minimum value. If \( a \) is negative, the parabola opens downwards, and the vertex gives the maximum value.

What is the minimum value of f(x) = 2x^2 - 6x + 5?

Learn how to determine the minimum value of the quadratic function f(x) = 2x^2 - 6x + 5. This problem involves using the vertex formula for parabolas, suitable for students in Grades 9-11. Detailed step-by-step solution included.

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