Final answer:
To determine the value of -4a, we solve a system of equations created by substituting x = 2 and x = 1 into the function f(x) = ax^3 + bx - 1. Through the solving process, we find that a = -1/4, hence -4a = 1.
Step-by-step explanation:
To find the value of -4a given the function f(x) = ax^3 + bx - 1, and knowing that f(2) = 3 and f(1) = 7/4, we need to substitute these x values into the given function and solve the system of equations they create.
First, substituting x = 2:
- 3 = a(2)^3 + b(2) - 1
- 3 = 8a + 2b - 1
- 4 = 8a + 2b
Second, substituting x = 1:
- 7/4 = a(1)^3 + b(1) - 1
- 7/4 = a + b - 1
- 11/4 = a + b
From these equations, we can solve for a and b.
4 = 8a + 2b (1)
11/4 = a + b (2)
Multiplying equation (2) by 2 gives:
11/2 = 2a + 2b
Subtracting this from equation (1) we get:
4 - 11/2 = 8a + 2b - (2a + 2b)
8/2 - 11/2 = 6a
-3/2 = 6a
a = -1/4
Finally, we find that -4a = -4(-1/4) = 1.