JEE PYQ: Application Of Derivatives Question 24
Question 24 - 2021 (26 Feb Shift 2)
Let $a$ be an integer such that all the real roots of the polynomial $2x^5 + 5x^4 + 10x^3 + 10x^2 + 10x + 10$ lie in the interval $(a, a+1)$. Then $|a|$ is equal to
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Answer: 2
Solution
$f(x) = 2x^5 + 5x^4 + 10x^3 + 10x^2 + 10x + 10$. $f’(x) = 10\left(x^2 + x + 1\right)^2 + \text{positive terms} > 0$ for all $x \in \mathbb{R}$, so $f$ is strictly increasing with exactly one real root. $f(-1) = 3 > 0$, $f(-2) = -34 < 0$. Root is in $(-2, -1)$, so $a = -2$, $|a| = 2$.
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