Let f : R R be defined as f ( x ) = a e 2 x + b e x + c x. If f ( 0 ) = - 1 , f ' ( log e 2 ) = 21 and 0 log e 4 ( f ( x ) - c x ) ? d x = 39 2 , then the value of | a + b + c | equals [2024]

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Solution

Q.
Let $f : R \to R$ be defined as $f (x) = a e^{2 x} + b e^{x} + c x .$ If $f (0) = - 1, f' (\log_{e} 2) = 21 and \int_{0}^{\log_{e} 4} (f (x) - c x) d x = \frac{39}{2},$ then the value of $| a + b + c |$ equals [2024]

1 12

2 16

3 8

4 10

Ans.
(3)

Given, $f (x) = a e^{2 x} + b e^{x} + c x$

$\Rightarrow f (0) = a (1) + b (1) + c (0)$

$\Rightarrow a + b = - 1$ ...(i)

$Also, f' (x) = 2 a e^{2 x} + b e^{x} + c$

$\Rightarrow f' (\log_{e} 2) = 8 a + 2 b + c = 21$ ...(ii)

$Now, \int_{0}^{\log_{e} 4} (f (x) - c x) d x$

$= \int_{0}^{\log_{e} 4} (a e^{2 x} + b e^{x}) d x = \frac{a}{2} (16 - 1) + b (4 - 1)$

$= \frac{15 a}{2} + 3 b = \frac{39}{2} = \frac{9 a}{2} + 3 (a + b) = \frac{39}{2}$

$\Rightarrow \frac{9 a}{2} = \frac{39}{2} + 3$ $(Since, a + b = 1)$

$\Rightarrow \frac{9 a}{2} = \frac{45}{2} \Rightarrow a = 5$

$From (i) and (ii), we get b = - 6 and c = - 7$

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