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Among several applications of maxima and minima is finding the largest term of a sequence. Let \( a_{n} \) be a sequence. Consider \( f(x) \) obtained by replacing \( x \) by \( n \) e.g. let \( a_{n}=\frac{n}{n+1} \) consider \( f(x)=\frac{x}{x+1} \) on \( [1, \infty) f^{\prime}(x)= \) \( \frac{1}{(x+1)^{2}}0 \) for all \( x \). Hence \( \max f(x)=\lim _{x \rightarrow \infty} f(x)=1 \), so the largest term of \( a_{n} \) is 1 . If \( f(x) \) is the function required to find largest term in \( a_{n}=n^{2} /\left(n^{3}+200\right) \) then
(a) \( f \) increases for all \( x \)
(b) \( f \) decreases for all \( x \)
(c) \( f \) has a maximum at \( x=\sqrt[3]{400} \)
(d) \( f \) increases on \( [0,9] \).
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