JEE Main 2022 · 29 Jun · Shift 2 · Mathematics Q6

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MathematicsSingle correct+4 / -1Question 6 of 90

Let $f:\mathbb R\to\mathbb R$ 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?

For

n_1=3

n_2=4

, there exists

\alpha\in(3,5)

where

f

attains local maxima.

For

n_1=4

n_2=3

, there exists

\alpha\in(3,5)

where

f

attains local minima.

For

n_1=3

n_2=5

, there exists

\alpha\in(3,5)

where

f

attains local maxima.

For

n_1=4

n_2=6

, there exists

\alpha\in(3,5)

where

f

attains local maxima.

Answer and worked solution

Official answer: C

$n_1=3$ , $n_2=5$ . On $(3,5)$ , $f=(x-3)^3(x-5)^5<0$ and vanishes at both ends, so the turning point there is a minimum, not a maximum.

Others hold: (A) $f>0$ inside gives a maximum; (B) $f<0$ gives a minimum; (D) $f>0$ gives a maximum.

(C)

Topics:Maxima and minima

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JEE Main 2022 · 29 Jun · Shift 2, Mathematics question 6