Let f (x) = ax5 + bx4 + cx3 + dx2 + ex, Where a, b, c, d, e Ïµ R And f (x) = 0 Has A Positive Root α, Then

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Question

Let f (x) = ax⁵ + bx⁴ + cx³ + dx² + ex, where a, b, c, d, e Ïµ R and f (x) = 0 has a positive root α, then

medium

Solution

Correct option is

All of the above

It is given that α is a positive root of f (x) and by inspection, we have f (0) = 0

∴ x = 0 and x = α are roots of f (x) = 0.

By Rolle’s theorem, f’(x) = 0 has a root α₁ between 0 and α i.e., 0 < α₁ < α

∴ (a) is correct.

Clearly, f’(x) = 0 is a fourth degree equation in x and imaginary roots always occurs in pairs.

Since x = α₁ is a root of f’(x) = 0

∴ f’(x) = 0 will have another real root, α₂ (say)

∴ Now, α₁ and α₂ are real roots of f’(x) = 0

∴ By Roll’s Theorem f’’ (x) = 0 will have a real root between α₁ and α₂.

∴ (b) is correct.

We have seen that x = 0, x = α are two real roots of f (x) = 0. As

f (x) = 0 is fifth degree equation, it will have at least three real roots. Consequently by Rolle’s

Theorem f’(x) = 0 will have at least two real roots.

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