Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value.
$f (x) = 3 x^{2} + 18 x - 1$
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.)

Answer

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Answer

Final Answer: The minimum value of the function is $- 28$ .

Steps

Step 1 ：The given function is a quadratic function of the form $f (x) = a x^{2} + b x + c$ .

Step 2 ：The coefficient of $x^{2}$ is positive, so the parabola opens upwards. This means that the function has a minimum value.

Step 3 ：The minimum or maximum value of a quadratic function $f (x) = a x^{2} + b x + c$ is given by $f (- \frac{b}{2 a})$ .

Step 4 ：So, we need to substitute $x = - \frac{b}{2 a}$ into the function to find the minimum value.

Step 5 ：Given that $a = 3$ , $b = 18$ , and $c = - 1$ , we find that $x = - 3.0$ .

Step 6 ：Substituting $x = - 3.0$ into the function, we find that the minimum value is $- 28.0$ .

Step 7 ：Final Answer: The minimum value of the function is $- 28$ .