Question

For some function f(x) we have f(a) = f(b) for ∀a,b…∊….I and a + b = 2 then which of the following even degree polynomials could f(x) be

(a) x^2 + 3x +1

(b) ^5x^2⁄2 – 5x + 101

(c) x^2 + 2x + 1

(d) Even degree polynomials of such kind cannot exist

I got this question in an interview.

The above asked question is from Rolle’s Theorem in portion Differential Calculus of Engineering Mathematics

Answer 1

The correct option is (b) ^5x^2⁄2 – 5x + 101

The explanation is: Given that f(a) = f(b) for all integral values of a, b with the condition \(\frac{a+b}{2}\) = 1

We know that the function is symmetric about the line x = 1

Thus we must have a Rolle point at 1 for any function that satisfies these conditions.

Differentiating the given functions in the options we have for a

2x + 3 = 0

x = ^-3⁄2 ≠ 1

Doing the same for (b) we get

5x – 5 = 0

x = 1.