Question

Let f (x) = (x – a) (x – b) (x – c), a < b < c. Then  f'(x) = 0 has two roots. At which interval does these roots belongs?

(a) Both the roots in (a, b)

(b) At least one root in (a, b) and at least one root in (b, c)

(c) Both the roots in (b, c)

(d) Neither in (a, b) nor in (b, c)

The question was posed to me during an interview for a job.

The question is from First Order Derivative in section Limits and Derivatives of Mathematics – Class 11

Answer 1

Right option is (b) At least one root in (a, b) and at least one root in (b, c)

The best explanation: Here, f (x) being a polynomial is continuous and differentiable for all real values of x.

We also have f(a) = f(b) = f(c). If we apply Rolle’s theorem to f (x) in [a, b] and [b, c] we will observe that f'(x) = 0 will have at least one root in (a, b) and at least one root in (b, c).

But f'(x) is a polynomial of degree two, so that f'(x) = 0 can’t have more than two roots. It implies that exactly one root of f'(x) = 0 will lie in (a, b) and exactly one root off'(x) = 0 will lie in (b, c).

Let y = f(x) be a polynomial function of degree n. If f (x) = 0 has real roots only, then f'(x) = 0, f”(x) = 0, … , f^n-1(x) = 0 will have real roots. It is in fact the general version of above mentioned application, because if f (x) = 0 have all real roots, then between two consecutive roots of f(x) = 0, exactly one root of f'(x) = 0 will lie.