HFined \( \int 3 x^{2} e^{2 x^{3}} d x \)

A. The critical number of \( f(x) \) B. Find the intervals on which \( f \) is increasing and decreasing by using the first derivative test Find the local maximum and minimum, if they exist by using the first derivative test Find the intervals on which \( f \) is concave up and concave down Find the inflection points of \( f(x) \)

20. \( \int x \tan ^{2} x d x \)

Which function satisfies the differential equation \( y^{\prime \prime}=-y \)

What is the solution to the following differential equation, subject to the initial condition \( y(0)=2 \) ? \( \frac{d y}{d x}=6 x+6 \)

Un objeto se mueve a lo largo de una línea y su aceleración está dada por \( a(t) = 12t \). Si en el tiempo inicial \( t=0 \) la velocidad es \( v(0)=4 \), encuentra la expresión para la velocidad.