Consider the function defined for

MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives

Problem

Consider the function

g(x)=\dfrac{x}{x^{3}+1}

defined for

x\ge 1

. The term

g(n)

appears in a sequence whose behavior you will analyze. (a) Calculate

g'(x)

. (b) Solve the inequality

g'(x)<0

for

x\ge 1

. (c) Determine the least positive integer

N

such that the sequence

\{g(n)\}

is strictly decreasing for all integers

n\ge N

. Provide clear justification for your choice of

N