Fit an equation to a max and a min?

Question

Fit a function in the form ax^3 + bx^2 + cx + d to a local minimum and maximum at (4,7) and (6,-2)

I know this involves a system of equations and substitution but I have no idea how to do it. Thanks!

Answer 1

dy/dx = 3ax^2 +2bx +c
7 = 48a + 8b +c <-- Eq 1
6 = 12a -4b +c <-- Eq 2
7 = 64a +16b +4c +d <-- Eq 3
6 = -8a +4b -2c +d <-- Eq 4
So you have 4 equations in 4 unknowns.

You could solve this on your caculator using matrices.
Or you could just start by subtracting Eq 4 from Eq 3 thereby eliminating d and now you wuld have just three equations in three unknowns.

Answer 2

First diffferinate equation and set to 0, and insert x, so
3a*4^2+2b*4+c=0, or 48a+8b+c=0, and
3a*6^2+2b*6+c=0, or 108a+12b+c=0.

And insert points in original equation, giving you
a*4^3+b*4^2+c4+d=7, or 64a+16b+4c+d=7
and a*6^3+b*6^2+6c+d=-2, or 216a+36b+6c+d=-2.
Using all the equations, I get a=9/4, b=-135/4, c=162, d=-245.