To find the maximum value of the function $f(x) = -2x^2 + 4x + 3$, we can proceed by completing the square or by using the vertex formula for a quadratic function in the form $ax^2 + bx + c$.

Step-by-Step Solution

The function $f(x) = -2x^2 + 4x + 3$ is a downward-opening parabola because the coefficient of $x^2$ is negative (i.e., $-2$).

1. Identify the coefficients:

   • $a = -2$
   • $b = 4$
   • $c = 3$

2. Find the vertex: For a parabola $y = ax^2 + bx + c$, the $x$-coordinate of the vertex is given by: $x = -\frac{b}{2a}$ Substituting the values of $a$ and $b$: $x = -\frac{4}{2 \times -2} = 1$

3. Calculate the maximum value: Substitute $x = 1$ back into $f(x)$ to find $f(1)$: $f(1) = -2(1)^2 + 4(1) + 3$ $f(1) = -2 + 4 + 3 = 5$

Conclusion

The maximum value of the function $f(x) = -2x^2 + 4x + 3$ is $\mathbf{5}$.

Would you like more details on any step, or have additional questions?

Here are some related questions to consider:

1. What are the steps for completing the square to find maximum or minimum values?
2. How does the sign of the coefficient of $x^2$ affect the shape of a parabola?
3. What is the significance of the vertex in a quadratic function?
4. How would the function change if the coefficient of $x^2$ were positive?
5. Can this method be used to find maximum or minimum values of higher-degree polynomials?

Tip: For any quadratic function $ax^2 + bx + c$, you can quickly determine if it has a maximum (downward parabola) or minimum (upward parabola) by checking the sign of $a$.