To find the minimum or maximum value of the function $f(x) = 2x^2 - 16x + 30$, we need to determine the vertex of this quadratic function, as the vertex represents the point of the minimum or maximum value.

Step 1: Identify the function's form

The given function is a quadratic function of the form:

$f(x) = ax^2 + bx + c$

where:

• $a = 2$
• $b = -16$
• $c = 30$

Since $a > 0$, the parabola opens upwards, indicating a minimum value at the vertex.

Step 2: Find the x-coordinate of the vertex

The x-coordinate of the vertex can be found using the formula:

$x = -\frac{b}{2a}$

Substituting the values of $a$ and $b$:

$x = -\frac{-16}{2 \times 2} = \frac{16}{4} = 4$

Step 3: Find the y-coordinate (minimum value)

To find the y-coordinate of the vertex, substitute $x = 4$ into the function:

$f(4) = 2(4)^2 - 16(4) + 30$

$f(4) = 2(16) - 64 + 30$

$f(4) = 32 - 64 + 30 = -2$

Conclusion

The function has a minimum value, and that value is:

$\text{Minimum value} = -2$

Would you like more details or have any questions? Here are some related questions that might interest you:

1. How do you determine if a quadratic function has a maximum or minimum value?
2. What are some real-world applications of finding maximum or minimum values of functions?
3. How would the minimum value change if the coefficients of the function were altered?
4. Can the minimum value of this function be verified graphically?
5. How does the discriminant relate to the minimum or maximum of a quadratic function?

Tip: For any quadratic function $ax^2 + bx + c$, if $a > 0$, the function has a minimum value; if $a < 0$, it has a maximum value.