Question

A cubic f(x) vanishes at x = −2 and has relative minimum/ maximum at x = −1 and x = 1/3. If ∫f(x)dx for x ∈ [-1, 1] = 14/3, the cubic f(x) = λ₁x³ + λ₂x² − x + 2, then find (λ₁ + λ₂).

Answer 1

x = − 1 and x = 1/3 are roots of f′(x) = 0. Therefore,

f′(x) = a(3x − 1) (x + 1) = a(3x² + 2x − 1)

⇒ f(x) = a(x³ + x² − x + b) f(−2) = 0

⇒ b = 2 ⇒ f(x) = a(x³ + x² − x + 2)

Therefore, 

f(x) = x³ + x² − x + 2