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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
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+1 vote

Answer: $C$

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

Substitute $x$ with $-x$, in the above question, we get:

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

Multiplying equation $(1)$ with $3$ and equation $(2)$ with $2$, we get:

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

and,

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

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

$5f(-x) = 15-20x$

Now, substitute, $x = -2$, we will get $f(2)$:

$5f(2) = 15 - 20*-2 \implies 5f(2) =55 \implies f(2) = \bf{11}$

$\therefore 11$ is the correct answer.

by Boss (18.9k points)
edited by