QJEEM19Ma4c5d: JEE Main 2019 Mathematics - Class 12 JEEM19Ma4c5d Solution: Let f be a differentiable function such that f '(x) = 7 - (x >…

Question

Verified Solution • 2025–26 Session moderate

Let f be a differentiable function such that f '(x) = 7 -

{3 \over 4}{{f\left( x \right)} \over x},

(x > 0) and f(1)

\ne

4. Then

\mathop {\lim }\limits_{x \to 0'} \,

xf

\left( {{1 \over x}} \right)

:

Accepted Answer

f'(x) = 7 - {3 \over 4}{{f\left( x \right)} \over x}\,\,\,\left( {x &gt; 0} \right)

Given f(1)

\ne

4

\mathop {\lim }\limits_{x \to {0^ + }} \,xf\left( {{1 \over x}} \right)\, = ?

{{dy} \over {dx}} + {3 \over 4}{y \over x} = 7

(This is LDE)

IF

= {e^{\int {{3 \over {4x}}dx} }} = {e^{{3 \over 4}\ln \left| x \right|}} = {x^{{3 \over 4}}}

y.{x^{{3 \over 4}}} = \int {7.{x^{{3 \over 4}}}} dx

y.{x^{{3 \over 4}}} = 7.{{{x^{{7 \over 4}}}} \over {{7 \over 4}}} + C

f(x) = 4x + C.{x^{ - {3 \over 4}}}

f\left( {{1 \over 4}} \right) = {4 \over x} + C.{x^{{3 \over 4}}}

\mathop {\lim }\limits_{x \to {0^ + }} xf\left( {{1 \over x}} \right) = \mathop {\lim }\limits_{x \to {0^ + }} \left( {4 + C.{x^{{7 \over 4}}}} \right) = 4

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