Let f be a function such that 3 f ( x ) + 2 f ( m 19 x ) = 5 x , x 0 , where m = i = 1 9 ( i ) 2 . Then f ( 5 ) - f ( 2 ) is equal to [2026]

Question

Let $f$ be a function such that $3 f (x) + 2 f (\frac{m}{19 x}) = 5 x, x \neq 0,$ where $m = \sum_{i = 1}^{9} {(i)}^{2} .$ Then $f (5) - f (2)$ is equal to [2026]

Smart Achievers · Accepted Answer

(1)

$m = \frac{9 \times 10 \times 19}{6} = 15 \times 19$

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

$Replace x by \frac{15}{x}$

$3 f (\frac{15}{x}) + 2 f (x) = \frac{75}{x}$

$9 f (x) - 4 f (x) = 15 x - \frac{150}{x}$

$5 f (x) = 15 x - \frac{150}{x}$

$f (x) = 3 x - \frac{30}{x}$

$f (5) = 15 - \frac{30}{5} = 9$

$f (2) = 6 - 15 = - 9$

$f (5) - f (2) = 18$

Solution

Q.
Let $f$ be a function such that $3 f (x) + 2 f (\frac{m}{19 x}) = 5 x, x \neq 0,$ where $m = \sum_{i = 1}^{9} {(i)}^{2} .$ Then $f (5) - f (2)$ is equal to [2026]

1 18

2 - 9

3 9

4 36

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