ISI2014-DCG-6

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If $f(x)$ is a real valued function such that $2f(x)+3f(-x)=15-4x$, for every $x \in \mathbb{R}$, then $f(2)$ is

1. $-15$
2. $22$
3. $11$
4. $0$
in Calculus
recategorized

Answer: $\mathbf C$

Solution:

Given: $\mathrm {2f(x) + 3f(-x) = 15 - 4x }\tag{1}$

Substitute $\color {red} {\mathrm x}$ with $\color {red} {\mathrm {-x}}$, in the above question, we get:

$\mathrm {2f(-x) + 3 f(x) = 15 +4 x} \tag {2}$

On multiplying, Equation $(1)$ $\times$ $3$ and Equation $(2)$ $\times$ $2$, we get:

$\mathrm {6f(x) + 9f(-x) = 45-12x} \tag {3}$

and,

$\mathrm {4f(-x) + 6f(x) = 30 + 8x} \Rightarrow \mathrm {6f(x) + 4f(-x) = 30+8x} \tag{4}$

Subtracting equation $(3)$ and $(4)$,we get:

$\mathrm {5f(-x) = 15-20x}$

Now, substitute, $\mathrm x = -2$, we get:

$\mathrm {5f(2) = [15 - 20\times (-2)]}$

$\Rightarrow \mathrm{5f(2)} = 55$

$\Rightarrow \mathrm{f(2) = \bf{11}}$

$\therefore \mathbf C$ is the correct option.

edited by

Given :                              2 f (x) + 3 f(-x) = 15 - 4x

put x = 2                            2 f (2) + 3 f(-2) =  7       ..................( 1 )

put x = -2                           2 f(-2)  + 3 (2) = 23       ...................( 2 )

Now    do       2 * eq. ( 1 ) - 3 * eq. ( 2 )

5 * f ( 2) = 55

f ( 2 )=11

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