the question states: Find the coefficients of the fourth degree polynomial:
p(x) = ax^4 + bx^3 + cx^2 + dx + e whose graph goes through the points (0,0), (1,1), (-1,3) and whose slope at x=-1 is 20 and x=1 is 9.
The Attempt at a Solution
i started by putting it into an augmented matrix, and solving...
my matrix was:
0 0 0 0 1 | 0
1 1 1 1 1 | 1
1 -1 1 -1 1 | 3
when i solved for the coefficients, i got a = 0, b = -1, c = 2, d = 0, and e = 0.
i'm not sure how to incorporate the information given by the slopes.
i know that you can take the derivative to find slope.. and i took the derivative of the polynomial and got:
4ax^3 + 3bx^2 + 2cx + d
but I'm not really sure where to go from there. do i plug in -1 for x and set it equal to 20 and 1 for x and set it equal to 9? if i do that i get the equations:
-4a + 3b - 2c + d = 20 and
4a + 3b + 2c + d = 9
by now i think I'm starting to make things up though!
can anyone offer any hints? thank you!