Questions: The function f(x)=-x^2-2x+24 has a maximum of

Solution Steps

To find the maximum or minimum of a quadratic function in the form \( f(x) = ax^2 + bx + c \), we can use the vertex formula. The vertex \( x \)-coordinate is given by \( x = -\frac{b}{2a} \). Since the coefficient of \( x^2 \) is negative, the parabola opens downwards, indicating a maximum point. We substitute this \( x \)-coordinate back into the function to find the maximum value.

The given function is \( f(x) = -x^2 - 2x + 24 \). This is a quadratic function of the form \( ax^2 + bx + c \), where \( a = -1 \), \( b = -2 \), and \( c = 24 \).

The vertex of a parabola given by \( ax^2 + bx + c \) is at \( x = -\frac{b}{2a} \). Substituting the values of \( a \) and \( b \):

\[ x = -\frac{-2}{2 \times -1} = -1.0 \]

Since the coefficient of \( x^2 \) is negative, the parabola opens downwards, indicating a maximum point at the vertex. Substitute \( x = -1.0 \) back into the function to find the maximum value:

\[ f(-1.0) = -(-1.0)^2 - 2(-1.0) + 24 = 25.0 \]

Final Answer