Question

What is the interval of f(x) = (x – 1)(x – 2)(x – 3)/(x^3 + 6x^2 + 11x+ 6) where f(x) is negative?

(a) (-∞, -3) ∪ (3, ∞)

(b) (3, -2) ∪ (1, 1) ∪ (2, 3)

(c) (-∞, -3) ∪ (2, -1) ∪ (1, 2) ∪ (3, ∞)

(d) (-∞, ∞)

I have been asked this question in exam.

My question is based upon Sets and their Representations in section Sets of Mathematics – Class 11

Answer 1

The correct option is (b) (3, -2) ∪ (1, 1) ∪ (2, 3)

Easy explanation: f(x) = (x – 1)(x – 2)(x – 3)/(x^3 + 6x^2 + 11x+ 6)

After solving the cubic equation (x^3 + 6x^2 + 11x+ 6) we get (x+1)(x+2)(x+3)

Now, we can see that this implies f(x) = (x – 1)(x – 2)(x – 3)/(x + 1)(x + 2)(x + 3)

So, the critical points of x are, x = 1, 2, 3, -1, -2, -3

So, for f(x) < 0 ᵾ x €  (3, -2) ∪ (1, 1) ∪ (2, 3).