JEE PYQ: Application Of Derivatives Question 46
Question 46 - 2020 (07 Jan Shift 2)
Let $f(x)$ be a polynomial of degree 5 such that $x = \pm 1$ are its critical points. If $\lim_{x \to 0}\left(2 + \frac{f(x)}{x^3}\right) = 4$, then which one of the following is not true?
(1) $f$ is an odd function
(2) $f(1) - 4f(-1) = 4$
(3) $x = 1$ is a point of maxima and $x = -1$ is a point of minima of $f$
(4) $x = 1$ is a point of minima and $x = -1$ is a point of maxima of $f$
Show Answer
Answer: (3)
Solution
$f(x) = ax^5 + bx^4 + cx^3$ with $2 + c = 4 \Rightarrow c = 2$. $f’(1) = 0$, $f’(-1) = 0$: $5a + 4b + 6 = 0$, $5a - 4b + 6 = 0$. So $b = 0$, $a = -\frac{6}{5}$. $f(x) = -\frac{6}{5}x^5 + 2x^3$. $f’’(x) = -6x^2(x^2-1)/(…)$. $f$ has minima at $x = -1$ and maxima at $x = 1$.
BETA
