Mathematics

Problem

f(x)=\int \frac{x}{2}\,d\left(\frac{x^2-1}{x^2}\right)

and

f(2)=\frac{1}{2}

and

g_1(x)=f\big(f(f(\dots f(x)\dots))\big)

i.e.,

g_1(x)=f(x)

g_2(x)=f(f(x))

and so on then identify the correct statement(s). (a)

\frac{d}{dx}\big(g_{3n-2}(x)\big)=1

whenever exists,

n\in\mathbb{N}

(b)

\frac{d}{dx}\big(g_{3n}(x)\big)=1

whenever exists,

n\in\mathbb{N}

(c)

\lim_{x\to 1}\frac{\sum_{r=1}^{100}(g_{3r}(x))^r+x-101}{x-1}=5050

(d) Slope of the tangent to the graph of the function

y=g_{80}(x)

x=\frac{1}{2}

4

Intuitive explanation

This problem chains together integration by parts, function composition, and iteration. First, you'll use integration by parts on the given integral—recognize that $d\left(\frac{x^2-1}{x^2}\right)$ is your differential, so integrate $\frac{x}{2}$ against it. Once you find $f(x)$ using the boundary condition $f(2)=\frac{1}{2}$ , the real insight is spotting a pattern: composing $f$ with itself repeatedly ( $g_n$ means applying $f$ exactly $n$ times) often reveals a fixed point, a cycle, or a closed form that lets you predict $g_n(x)$ without computing every iteration.

Intuitive explanation

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