Answer:
To determine whether the function \(f(x) = -3x^2 + 18x - 25\) has a minimum, you can analyze its vertex. A quadratic function \(ax^2 + bx + c\) has a minimum when the coefficient \(a\) is negative, and the vertex of the parabola is the minimum point.
In this case, \(a = -3\), which is indeed negative, so the function is a downward-facing parabola, and it has a minimum. To find the x-coordinate of the vertex (the value of \(x\) at the minimum), you can use the formula:
\[x_{\text{min}} = \frac{-b}{2a}\]
For your function \(f(x) = -3x^2 + 18x - 25\), \(a = -3\) and \(b = 18\), so:
\[x_{\text{min}} = \frac{-18}{2(-3)} = \frac{-18}{-6} = 3\]
Now that you have the x-coordinate of the minimum point, you can find the corresponding y-coordinate by plugging \(x = 3\) into the function:
\[f(3) = -3(3)^2 + 18(3) - 25\]
\[f(3) = -3(9) + 54 - 25\]
\[f(3) = -27 + 54 - 25\]
\[f(3) = 54 - 52\]
\[f(3) = 2\]
So, the function \(f(x) = -3x^2 + 18x - 25\) has a minimum at \(x = 3\) and \(f(3) = 2\). The minimum value of the function is 2, and it occurs at \(x = 3\).
Explanation: