(B) \( F(x)=e^{2 x} \cdot\left(4 x^{5}-3\right)^{2} \)

Did you know that the function \( F(x) = e^{2x} \cdot (4x^{5} - 3)^{2} \) combines exponential and polynomial elements to create a fascinating example of a composite function? The exponential part, \( e^{2x} \), grows rapidly as \( x \) increases, while the polynomial part can oscillate based on the value of \( x \). This interplay can lead to interesting behaviors, such as points of inflection and local extrema! When working with a function like this, a common mistake is neglecting the chain rule when calculating its derivative. Remember, you’ll need to apply both the product rule and the chain rule to differentiate correctly. It's easy to get tangled up in layers of functions, so take it step by step! Factor out the exponential and treat the polynomial separately for a smoother ride through your calculations.