Question

the function f(x) = ax³+bx²+cx+d has three positive roots. If the sum of the roots of f(x) is 4,the largest possible integral value of c/a is .​

Answer 1

f(x) = ax³+bx²+cx+d = 0

let p, q and r the roots/zeros

pq + qr + rp = $\frac{c}{a}$

but,

pq +  qr + rp  ≤ p² + q² + r²

∴ p² + q² + r² = (p + q + r)² - 2(pq +  qr + rp)

                     = 16 - 2∑pq

∴ pq + qr + rp ≤ p² + q² + r²

⇒  pq + qr + rp ≤ 16 - 2(pq + qr + rp)

⇒ 3(pq + qr + rp) ≤ 16

⇒ pq + qr + rp ≤ $\frac{16}{3}$

⇒  $\frac{c}{a}\leq \frac{16}{3}$

∴ $\frac{c}{a}\leq 5.3$ Ans

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Answer 2

Step-by-step explanation:

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