Question 3 

If function f(x) is f(x)= -3x^2+2x+1, which point does constitute a maximum or minimum of the function? 

To determine whether the function \( f(x) = -3x^2 + 2x + 1 \) has a maximum or minimum, we first need to find the vertex of the parabola described by the quadratic function. 

### Step 1: Identify the coefficients
The function is in the standard form of a quadratic equation, \( f(x) = ax^2 + bx + c \), where:
- \( a = -3 \)
- \( b = 2 \)
- \( c = 1 \)

### Step 2: Calculate the vertex
The x-coordinate of the vertex (which gives the maximum or minimum point) can be found using the formula:

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

Substituting the values of \( a \) and \( b \):

\[
x = -\frac{2}{2(-3)} = \frac{2}{6} = \frac{1}{3}
\]

### Step 3: Find the y-coordinate of the vertex
To find the corresponding y-coordinate (which gives the value of \( f(x) \) at the vertex), substitute \( x = \frac{1}{3} \) back into the function:

\[
f\left(\frac{1}{3}\right) = -3\left(\frac{1}{3}\right)^2 + 2\left(\frac{1}{3}\right) + 1
\]

\[
f\left(\frac{1}{3}\right) = -3\left(\frac{1}{9}\right) + \frac{2}{3} + 1 = -\frac{1}{3} + \frac{2}{3} + 1 = \frac{1}{3} + 1 = \frac{4}{3}
\]

### Step 4: Determine if it's a maximum or minimum
Since the coefficient of \( x^2 \) (which is \( a = -3 \)) is negative, the parabola opens downwards. This means the vertex represents a **maximum** point.

### Conclusion
The function \( f(x) = -3x^2 + 2x + 1 \) has a maximum point at \( \left(\frac{1}{3}, \frac{4}{3}\right) \).

---

Do you want more details on this process or have any questions?

Here are 5 related questions you might find interesting:
1. How do you find the roots of the quadratic function \( f(x) = -3x^2 + 2x + 1 \)?
2. What is the axis of symmetry for the function \( f(x) = -3x^2 + 2x + 1 \)?
3. How would the vertex change if the function were \( f(x) = 3x^2 + 2x + 1 \)?
4. How do you determine the nature of the roots of a quadratic function using the discriminant?
5. Can you find the minimum or maximum value of a quadratic function without completing the square?

**Tip:** For any quadratic function \( f(x) = ax^2 + bx + c \), the sign of \( a \) determines whether the vertex is a maximum (if \( a < 0 \)) or a minimum (if \( a > 0 \)).

Discover how to find the maximum or minimum point of the quadratic function f(x) = -3x^2 + 2x + 1. Learn the vertex formula and how to determine whether a vertex represents a maximum or minimum point. Suitable for high school students studying quadratic functions.

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