Question

Let $f: R \rightarrow R$ 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 $\underline{\text { NOT }}$ true?

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

Answer 1

Answer: (C)

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