Question

Find the cubic function of the form y=ax^3 + bx^2 + cx + d which has a relative maximum point at (0,2) and a point of inflection at (-1,-2). How do I even start this one?

Answer 1

take the derivative, set it equal zero, and solve

Answer 2

then
\[y''=6ax+2b\] and you want this to be zero when x = -1, so you get
\[6a(-1)+2b=0\]
\[-6a+2b=0\]

Answer 3

ok i don't know why i deleted my reply, because it was right so i write it again.
you know
\[y'=3ax^2+2bx+c=0\] if x = 0
that tells you
\[c=0\]

Answer 4

and we also know that if x = 0, then y = 2 which tells you
\[d=2\]

Answer 5

so we have
\[y=ax^3+bx^2+2\]

Answer 6

and now use that if x = -1, y = -2 to write
\[-a+b+2=-2\]
\[-a+b=-4\] and solve the system
\[-a+b=-4\]
\[-6a+2b=0\] for "a" and "b"