Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined by $f(x)=(x-3)^{n_{1}}(x-5)^{n_{2}}, n_{1}, n_{2} \in \mathbb{N}$. Then, which of the following is NOT true?

(C) For $n_{1}=3, n_{2}=5$, there exists $\alpha \in(3,5)$ where $f$ attains local maxima.

(A) For $n_{1}=3, n_{2}=4$, there exists $\alpha \in(3,5)$ where $f$ attains local maxima.

(B) For $n_{1}=4, n_{2}=3$, there exists $\alpha \in(3,5)$ where $f$ attains local minima.

(D) For $n_{1}=4, n_{2}=6$, there exists $\alpha \in(3,5)$ where $f$ attains local maxima.

JEE Main · 2022 · 29 Jun 2022 · Shift 2 · Application of Derivatives

Let

f : R \to R

be a function defined by

f (x) = (x - 3)^{n_{1}} (x - 5)^{n_{2}}, n_{1}, n_{2} \in N

. Then, which of the following is NOT true?

(A)For $n_{1} = 3, n_{2} = 4$ , there exists $α \in (3, 5)$ where $f$ attains local maxima.
(B)For $n_{1} = 4, n_{2} = 3$ , there exists $α \in (3, 5)$ where $f$ attains local minima.
(C)For $n_{1} = 3, n_{2} = 5$ , there exists $α \in (3, 5)$ where $f$ attains local maxima.✓ Correct
(D)For $n_{1} = 4, n_{2} = 6$ , there exists $α \in (3, 5)$ where $f$ attains local maxima.

Answer: (C)

Compute

f^{'} (x) = (x - 3)^{n_{1} - 1} (x - 5)^{n_{2} - 1} [(n_{1} + n_{2}) x - (5 n_{1} + 3 n_{2})]

. The critical point in

(3, 5)

is at

x = \frac{5 n _{1} + 3 n _{2}}{n _{1} + n _{2}}

. For option (C) with

n_{1} = 3, n_{2} = 5

x = \frac{15 + 15}{8} = \frac{30}{8} = 3.75

. At this point,

f^{'} (x) = 8 (x - 3)^{2} (x - 5)^{4} (x - 3.75)

. The sign of

f^{'}

changes from negative to positive as

x

passes through 3.75, indicating a local minimum, not a maximum. Therefore, option (C) is NOT true.

JEE Main · 2022 · 29 Jun 2022 · Shift 2 · Single correct option