Consider the function $f(x)=-3 x^{2}+12 x-3$
a. Determine, without graphing, whether the function has a minimum value or a maximum value.
b. Find the minimum or maximum value and determine where it occurs.
c. Identify the function's domain and its range.
a. The function has a value.
Answer
\(\boxed{\text{Final Answer:}}\) The function \(f(x)=-3 x^{2}+12 x-3\) has a maximum value of 9 at \(x = 2\). The domain of the function is all real numbers and the range is \(y \leq 9\).
Step 1 ：The function given is \(f(x)=-3 x^{2}+12 x-3\).
Step 2 ：Since the coefficient of \(x^{2}\) is negative, the function opens downwards. Therefore, the function has a maximum value.
Step 3 ：The maximum value of a quadratic function \(f(x) = ax^{2} + bx + c\) occurs at \(x = -\frac{b}{2a}\).
Step 4 ：Substituting \(a = -3\) and \(b = 12\) into the formula, we get \(x = -\frac{12}{2(-3)} = 2\).
Step 5 ：The maximum value of the function is \(f(2) = -3(2)^{2} + 12(2) - 3 = 9\).
Step 6 ：The domain of the function is all real numbers, because any real number can be substituted for \(x\) in the function.
Step 7 ：The range of the function is \(y \leq 9\), because the maximum value of the function is 9 and the function opens downwards.
Step 8 ：\(\boxed{\text{Final Answer:}}\) The function \(f(x)=-3 x^{2}+12 x-3\) has a maximum value of 9 at \(x = 2\). The domain of the function is all real numbers and the range is \(y \leq 9\).
