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: $f(x) = \frac{1}{2}$.

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.