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AP EAPCET 25-Aug-2021 Shift 2 Paper

Section: Mathematics

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Question : 68 of 160

Marks: +1, -0

If

3 f(x)-2 f\left(\frac{1}{x}\right)=x

f'(2)=

$\frac{1}{2}$
$-\frac{1}{2}$
1
-1

Solution: 👈: Video Solution

3 f(x)-2 f\left(\frac{1}{x}\right)=x

...(i)

Replace

x

with

\frac{1}{x}

now.

From Eq. (i), we get

⇒

3 f\left(\frac{1}{x}\right)-2 f(x)=\frac{1}{x}

...(ii)

Eq. (i)

\times 2 +

Eq. (ii)

\times 3

\{6 f(x)-4 f\left(\frac{1}{x}\right)=2 x\}

+\{-6 f(x)+9 f\left(\frac{1}{x}\right)\}=2 x+\frac{3}{x}

5 f\left(\frac{1}{x}\right)=2 x+\frac{3}{x}

f\left(\frac{1}{x}\right)=\frac{2 x}{5}+\frac{3}{5 x}

∴

3 f(x)=x+2 f\left(\frac{1}{x}\right)

=x+2\left(\frac{2 x}{5}+\frac{3}{5 x}\right)

3 f(x)=x+\frac{4 x}{5}+\frac{6}{5 x}

⇒

f(x)=\frac{x}{3}+\frac{4 x}{15}+\frac{2}{5 x}

∴

f'(x)=\frac{1}{3}+\frac{4}{15}-\frac{2}{5 x^{2}}

f'(2)=\frac{1}{3}+\frac{4}{15}-\frac{2}{5(2)^{2}}

=\frac{1}{3}+\frac{4}{15}-\frac{1}{10}

=\frac{10+8-3}{30}=\frac{15}{30}

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

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