Find the absolute extrema of the function, if they exist, over the indicated interval. Also indicate the $x$-value at which each extremum occurs. If no interval is specified, use the real numbers, $(-\infty, \infty)$. \[ f(x)=0.003 x^{2}+4.2 x-90 \] Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The absolute maximum is $\square$ at $x=\square$ and the absolute minimum is $\square$ at $x=\square$. (Use a comma to separate answers as needed.) B. The absolute minimum is $\square$ at $x=\square$ and there is no absolute maximum. (Use a comma to separate answers as needed.) C. The absolute maximum is $\square$ at $x=\square$ and there is no absolute minimum. (Use a comma to separate answers as needed.) D. There is no absolute maximum and no absolute minimum.
Solution
Step 1 ：The function given is a quadratic function, which is a parabola. The coefficient of \(x^2\) is positive, so the parabola opens upwards. This means that the function has a minimum value, but no maximum value.
Step 2 ：The minimum value occurs at the vertex of the parabola, which can be found using the formula \(x = -\frac{b}{2a}\), where \(a\) is the coefficient of \(x^2\) and \(b\) is the coefficient of \(x\). In this case, \(a = 0.003\) and \(b = 4.2\).
Step 3 ：Substituting the values of \(a\) and \(b\) into the formula, we find that the \(x\)-value of the vertex is \(-700\).
Step 4 ：Substituting \(x = -700\) into the function, we find that the minimum value of the function is \(-1560\).
Step 5 ：\(\boxed{\text{The absolute minimum is } -1560 \text{ at } x=-700 \text{ and there is no absolute maximum.}}\)
