Note: Must show work to get full credit!
1. (10 points) Find the absolute maximum and minimum of $f(x)=x^{3}+3 x^{2}-1$ in $[-3,3]$.

Step 1 ：First, we find the derivative of the function \(f(x)=x^{3}+3 x^{2}-1\), which is \(f'(x)=3x^{2}+6x\).

Step 2 ：Next, we set the derivative equal to zero to find the critical points: \(3x^{2}+6x=0\). Solving this equation gives us the critical points \(-2\) and \(0\).

Step 3 ：We then evaluate the function at the critical points and the endpoints of the interval \([-3,3]\). The values we get are \(3\), \(-1\), \(-1\), and \(53\).

Step 4 ：Comparing these values, we find that the highest value is \(53\), which is the absolute maximum, and the lowest value is \(-1\), which is the absolute minimum.

Step 5 ：Final Answer: The absolute maximum of \(f(x)=x^{3}+3 x^{2}-1\) in \([-3,3]\) is \(\boxed{53}\) and the absolute minimum is \(\boxed{-1}\).