Consider 𝑝(𝑠) = 𝑠3 + 𝑎2𝑠2 + 𝑎1𝑠 + 𝑎0 with all real coefficients It is known that its derivative 𝑝′(𝑠) has no real roots

Q. Consider 𝑝(𝑠) = 𝑠³ + 𝑎₂𝑠² + 𝑎₁𝑠 + 𝑎₀ with all real coefficients. It is known that its derivative 𝑝′(𝑠) has no real roots. The number of real roots of 𝑝(𝑠) is

(A) 0                          (B) 1                          (C) 2                          (D) 3

Ans: 1

Sol:

Given p(s) = s3 + a2 s2 + a1s + a0

p'(s) = 3s2 + 2a2s + a1

Also given that p'(s) has no real roots, therefore, we can write:

n – 1 = 0

n = Number of real roots in p(s)

∴ n = 1