differential calculus (pls help) rush

Question

Determine the coefficients a, b, c, etc. so that the curve will satisfy the stipulated condition.

make the curve y = ax^4+bx^3+cx^2+dx+e pass though the points (1, 9) and have critical points at (0, 0) and (-1, -1).

Answer 1

y(0) = e = 0; y(- 1) = a - b + c - d = -1; y(1) = a + b + c + d = 9; y'(x) = 4ax³ + 3bx² + 2cx + d;

y'(0) = d = 0; y'(- 1) = - 4a + 3b - 2c = 0; 2a + 2c = 8; a + c = 4; 2b = 10; b = 5;

4a + 2c = 15; 2a = 7; a = 7/2; c = 1/2

y = 7/2x⁴ + 5x³ + 1/2x²