What is the maximum number of possible extreme values for the function, \( f(x)=x^{3}+4 x^{2}-3 x-18 \) ? A. 4 B. 1 c. 3 D. 2

To determine the maximum number of possible extreme values of the function \( f(x)=x^{3}+4 x^{2}-3 x-18 \), we first need to find its derivative, which will help us locate critical points. The derivative is \( f'(x) = 3x^2 + 8x - 3 \). Since this is a quadratic function, it can have at most two real roots. This means there can be up to two critical points where extreme values occur. However, each critical point could correspond to either a local maximum or minimum, giving us a maximum of two extreme values for a cubic function like this. Therefore, the answer is \( \text{D. 2} \). Cubic functions can have a range of behaviors, but when looking for extreme values, keep in mind that they can appear at critical points determined by the first derivative, and with a maximum of two critical points, one can estimate the maximum extremity effectively!