Question

Which of the following is correct for the nature of the roots x^5 – a0x^4 + 3ax^3 + bx^2 + cx + d = 0 if it is given that 2a0^2 < 15 and a0, a, b, c, d are real?

(a) Can’t be real

(b) Equal

(c) Real

(d) Depends on the value of x

The question was posed to me by my school teacher while I was bunking the class.

The origin of the question is Second Order Derivative topic in chapter Limits and Derivatives of Mathematics – Class 11

Answer 1

Right option is (a) Can’t be real

Easiest explanation: Let f(x) = x^5 – a0x^4 + 3ax^3 + bx^2 + cx + d so that,

Now, f’(x) = 5x^5 – 4a0x^3 + 9ax^2 + 2bx + c

And, f”(x) = 20x^3 – 12a0x^2 + 18ax + 2b

And, f”(x) = 60x^2 – 24a0x + 18a

= 6(10x^2 – 4a0x + 3a)

Now, discriminant of 10x^2 – 4a0x + 3a = 16a0^2 – 4

And it is given 8(2a0^2 – 15a) < 0

Hence, root of f”’(x) can’t be real

=>all the roots can’t be real.