uppose the function fsxd 5 50 2 9x 1 x4
y30
describes a physical situation that makes sense 
only for whole numbers between 0 and 10. For 
what value of x does f reach a minimum, and what 
is the minimum value? (Suggestion: We suggest 
beginning with a table starting at 0 with a table 
increment of 1 and then panning further down 
the table.)

It seems like the function you're referring to is incomplete or has some typographical errors. However, I can reconstruct the problem and solve it step by step. Based on what you’ve written, I believe the function might be similar to this:

\[ f(x) = 5x^2 - 50x + 2 \]

This is a quadratic function, which typically represents a parabola. Since the coefficient of \(x^2\) is positive (5), the parabola opens upwards, meaning that it has a minimum point.

Let's solve for the value of \(x\) where \(f(x)\) reaches a minimum and compute the corresponding minimum value.

### Step 1: Find the vertex of the parabola

For a quadratic function in the form:

\[
f(x) = ax^2 + bx + c
\]

The \(x\)-coordinate of the vertex, which gives the minimum (or maximum) point for the parabola, is found using the formula:

\[
x = -\frac{b}{2a}
\]

For our function \(f(x) = 5x^2 - 50x + 2\), we have:
- \(a = 5\)
- \(b = -50\)

Substitute these values into the formula:

\[
x = -\frac{-50}{2(5)} = \frac{50}{10} = 5
\]

Thus, the value of \(x\) where the function reaches its minimum is \(x = 5\).

### Step 2: Find the minimum value of the function

Now, substitute \(x = 5\) back into the original function to find the minimum value:

\[
f(5) = 5(5)^2 - 50(5) + 2
\]
\[
f(5) = 5(25) - 250 + 2
\]
\[
f(5) = 125 - 250 + 2 = -123
\]

### Conclusion:
- The function reaches its minimum value at \(x = 5\).
- The minimum value of the function is \(-123\).

Would you like more details or have any questions about this solution?

---

Here are 5 related questions for further exploration:
1. What does the vertex of a quadratic function represent in real-world problems?
2. How can we determine if a quadratic function opens upwards or downwards?
3. What is the axis of symmetry in a quadratic function, and how is it related to the vertex?
4. How can the discriminant be used to analyze the nature of the roots of a quadratic equation?
5. What other methods can be used to find the minimum value of a quadratic function?

**Tip**: For any quadratic function, the vertex is always the critical point, and its coordinates can be found easily using \(-\frac{b}{2a}\). This gives you valuable insights into the function's behavior.

Suppose the function fsxd 5 50 2 9x 1 x4 y30 describes a physical situation that makes sense only for whole numbers between 0 and 10. For what value of x does f reach a minimum, and what is the minimum value? (Suggestion: We suggest beginning with a table starting at 0 with a table increment of 1 and then panning further down the table.)

Discover how to find the minimum value of the quadratic function f(x) = 5x^2 - 50x + 2. This problem walks through the step-by-step process using the vertex formula and quadratic concepts. Perfect for students in Grades 9-10 exploring optimization in algebra.

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