Consider the function $f (x) = - 3 x^{2} + 30 x - 6$
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.

Answer

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Answer

The smallest possible value of $c$ is $5$ .

Steps

Step 1 ：We can complete the square, to get $f (x) = 96 - 3 (x - 5)^{2}$ .

Step 2 ：Thus, the graph of $f (x)$ is a parabola with axis of symmetry $x = 5$ , so the function has a maximum value.

Step 3 ：The maximum value of the function is $96$ and it occurs at $x = 5$ .

Step 4 ：The function's domain is all real numbers, and its range is \(-\infty, 96\].

Step 5 ：The smallest possible value of $c$ is $5$ .