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(C) The derivative of the function is given by $f'(x) = n_{1}(x-3)^{n_{1}-1}(x-5)^{n_{2}} + n_{2}(x-3)^{n_{1}}(x-5)^{n_{2}-1}$.Simplifying this,we get

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

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(C) The derivative of the function is given by $f'(x) = n_{1}(x-3)^{n_{1}-1}(x-5)^{n_{2}} + n_{2}(x-3)^{n_{1}}(x-5)^{n_{2}-1}$.Simplifying this,we get $f'(x) = (x-3)^{n_{1}-1}(x-5)^{n_{2}-1} [n_{1}(x-5) + n_{2}(x-3)] = (x-3)^{n_{1}-1}(x-5)^{n_{2}-1} [(n_{1}+n_{2})x - (5n_{1}+3n_{2})]$.The critical point in $(3, 5)$ is $x = \frac{5n_{1}+3n_{2}}{n_{1}+n_{2}}$.For $n_{1}=3, n_{2}=5$,$f'(x) = (x-3)^{2}(x-5)^{4} [8x - 30] = 8(x-3)^{2}(x-5)^{4} (x - 3.75)$.Since $(x-3)^{2}$ and $(x-5)^{4}$ are always non-negative,the sign of $f'(x)$ changes from negative to positive at $x = 3.75$,which implies a local minimum,not a maximum. Thus,option $C$ is not true.

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

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