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I.S.I B.STAT 2018 | SUBJECTIVE -4

PROBLEM

Let f (0,\infty)\rightarrow \mathbb{R} be a continous function such that for all x \in (0,\infty) f(x)=f(3x) Define g(x)= \int_{x}^{3x} \frac{f(t)}{t}dt for x \in (0,\infty) is a constant function

HINT

Use leibniz rule for differentiation under integral sign

SOLUTION

using leibniz rule for differentiation under integral sign we get
g'(x)=f(3x)-f(x)

\Rightarrow g'(x)=0 [ Because f(3x)=f(x)]
Since the derivative of g(x) is 0 for all x, Hence g(x) is a constant function

PROBLEM

Let f (0,\infty)\rightarrow \mathbb{R} be a continous function such that for all x \in (0,\infty) f(x)=f(3x) Define g(x)= \int_{x}^{3x} \frac{f(t)}{t}dt for x \in (0,\infty) is a constant function

HINT

Use leibniz rule for differentiation under integral sign

SOLUTION

using leibniz rule for differentiation under integral sign we get
g'(x)=f(3x)-f(x)

\Rightarrow g'(x)=0 [ Because f(3x)=f(x)]
Since the derivative of g(x) is 0 for all x, Hence g(x) is a constant function

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