Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value. "f(x) $$ =-2 x^{2}+4 x-7 $$ Does the quadratic function f have a minimum value or a maximum value? The function $$ f $$ has a minimum value. The function $$ f $$ has a maximum value. What is this minimum or maximum value? Simplify your answer.)

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1. The given quadratic function is  
   $$
   f(x) = -2x^2 + 4x - 7.
   $$

2. Since the coefficient of $$ x^2 $$ is $$-2$$ (which is negative), the parabola opens downward. This implies that the function has a maximum value.

3. The $$ x $$-coordinate of the vertex for a quadratic function $$ ax^2 + bx + c $$ is given by  
   $$
   x = -\frac{b}{2a}.
   $$
   Here, $$ a = -2 $$ and $$ b = 4 $$, so  
   $$
   x = -\frac{4}{2(-2)} = -\frac{4}{-4} = 1.
   $$

4. Substitute $$ x = 1 $$ back into the function to find the maximum value:  
   $$
   f(1) = -2(1)^2 + 4(1) - 7 = -2 + 4 - 7 = -5.
   $$

5. Therefore, the quadratic function has a maximum value, and this maximum value is  
   $$
   -5.
   $$
The quadratic function $$ f(x) = -2x^2 + 4x - 7 $$ has a maximum value of $$-5$$.

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