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 $\left(\frac{1}{3}, \frac{4}{3}\right)$.

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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 $a > 0$).