• Find dots. *f* is zero only when *x* = 0. We find an interesting new twist here, because *f* has asymptotes:
there's a vertical asymptote at *x* = 4 and a horizontal asymptote at *y* = 12. The derivative of *f* is This is only zero or undefined when *x* = 4. Since *f* is undefined when *x* = 4, we don't find a critical point here, so *f* has no critical points. The second derivative of *f* is Again, the only place *f *" is zero or undefined is at *x* = 4, where *f* is undefined. Therefore *f* has no inflection points either. When we go to graph the dots, we also put on the asymptotes: • Find shapes. Make the numberlines: The function is negative when 0 < *x* < 4 (since then 12*x* is negative and ( *x* - 4 ) is positive), and positive everywhere else: The derivative *f *'(*x*) = -48( *x* - 4 )^{2} is always negative, since ( *x* – 4 )^{2} is always positive. The second derivative is negative when ( *x* – 4 ) is negative and positive when ( *x* – 4 ) is positive: By looking at *f *' and *f *" we can figure out the shapes of *f *: • Play connect-the-dots. We connect the dots to find the final graph, remembering to use the asymptotes: |