Answer:
(a, b, c, d) = (2, -4, 0, -6)
Explanation:
You want the coefficients in f(x) = ax^4 +bx^2 +cx +d such that (0, -6) is a maximum and (1, -8) is a minimum.
Function values
The given points help you write two equations in the four unknown values.
f(0) = -6
a·0 +b·0 +c·0 +d = -6 ⇒ d = -6
f(1) = -8
a·1 +b·1 +c·1 -6 = -8 ⇒ a +b +c = -2
Derivative values
The derivative of the function is ...
f'(x) = 4ax^3 +2bx +c
At the given extremes, the derivative is zero, so we have ...
4a·0 +2b·0 +c = 0 ⇒ c = 0
4a·1 +2b·1 +0 = 0 ⇒ 2a +b = 0
Solution
Using the equations we have, we can solve for 'a' and 'b':
Subtracting the first equation from the second, we get ...
(2a +b) -(a +b) = (0) -(-2)
a = 2
The first equation gives the value of b:
b = -2 -a = -2 -2
b = -4
The values of a, b, c, d are ...
(a, b, c, d) = (2, -4, 0, -6)