Question

For what values of a is the function f(x) = ((a² – 1)/3)x³ + (a – 1)x² + 2x + 1 strictly increasing?

Answer 1

Given f(x) = ((a² – 1)/3)x³ + (a – 1)x² + 2x + 1

⇒ f'(x) = 3((a² – 1)/3)x² + 2(a – 1)x + 2

⇒ f'(x) = (a² – 1)x² + 2(a – 1)x + 2

Since f is strictly increasing, so f'(x) > 0

⇒ (a² – 1)x² + 2(a – 1) x + 2 > 0

⇒ (a² – 1) > 0 & 4(a – 1)² – 8(a² – 1) < 0

⇒ a² > 1 & (a – 1)² – 2(a² – 1) < 0

⇒ (a + 1)(a – 1) > 0 & (a + 3)(a – 1) > 0

⇒ a ∈ (–∞, –1) ∪ (1, ∞) and a ∈ (–∞, –3) ∪ (1, ∞)

Hence, the values of a are a ∈(–∞, –3) ∪ (1, ∞)