Suppose the function f(x)=2x^3+ax^2+bx has a critical point at x=−1, for which f(−1)=−2. Determine a, and b?
f(-1) = -2, so
2(-1)^3+a(-1)^2+b(-1) = -2
-2 + a - b = -2
a - b = 0 --------- (1)
differentiating, f' = 6x^2 + 2ax + b
f'(-1) = 6(-1)^2 + 2a(-1) + b = 0 (we have a critical point at x = -1, so f' is zero).
6 - 2a + b = 0
2a - b = 6 -------- (2)
Comparing (1) and (2), a = 6, b = 6