Let f: R arrow R be a function defined by f( x )=( x -3) n 1( x -5) n 2, n 1, n 2 ∈ N. The, which of the following is underline text NOT true?

Question

Let f: R arrow R be a function defined by f( x )=( x -3) n 1( x -5) n 2, n 1, n 2 ∈ N. The, which of the following is underline text NOT true? (A) For n 1=3, n 2=4, there exists α ∈(3,5) where f attains local maxima. (B) For n 1=4, n 2=3, there exists α ∈(3,5) where f attains local manima. (C) For n 1=3, n 2=5, there exists α ∈(3,5) where f attains local maxima. (D) For n 1=4, n 2=6, there exists α ∈(3,5) where f attains local maxima.

Tardigrade Admin · Accepted Answer

f prime(x)=(x-3)n1-1(x-5)n2-1(n1+n2)(x-(5 n1+3 n2/n1+n2)) Option (3) is incorrect since for n 1=3, n 2=5 f prime(x)=8(x-3)2(x-5)4(x-(30/8)) minima at x=(30/8)

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

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

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

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

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

$f^{'} (x) = (x - 3)^{n_{1} - 1} (x - 5)^{n_{2} - 1} (n_{1} + n_{2}) (x - \frac{5 n _{1} + 3 n _{2}}{n _{1} + n _{2}})$
Option (3) is incorrect since
for $n_{1} = 3, n_{2} = 5$
$f^{'} (x) = 8 (x - 3)^{2} (x - 5)^{4} (x - \frac{30}{8})$
minima at $x = \frac{30}{8}$

Q. Let f:R→R be a function defined by f(x)=(x−3)n1​(x−5)n2​,n1​,n2​∈N. The, which of the following is NOT ​ true?

For n1​=3,n2​=4, there exists α∈(3,5) where f attains local maxima.

For n1​=4,n2​=3, there exists α∈(3,5) where f attains local manima.

For n1​=3,n2​=5, there exists α∈(3,5) where f attains local maxima.

For n1​=4,n2​=6, there exists α∈(3,5) where f attains local maxima.