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Question

If $$f(x)$$ is a polynomial in $$x(>0)$$ satisfying the equation $$ f(x)+f\left (\dfrac1x\right)=f(x).f\left (\dfrac1x\right)$$ and $$f(2)=-7$$,  then $$f(3)$$ is equal to


A
26
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B
27
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C
28
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D
29
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Solution

The correct option is A $$-26$$
Whenever, $$f(x)+f\left (\dfrac{1}{x}\right)=f(x).f\left (\dfrac{1}{x}\right)$$, then $$f(x)=1+x^n$$ or $$f(x)=1-x^n$$ keeping in mind that $$n$$ must be greater than $$0$$.
So here, $$f(2)=-7$$.
So we choose $$f(x)=1-x^n$$.
So, $$f(2)=1-2^n$$
$$\Rightarrow 2^n=8$$
$$\Rightarrow n=3$$
So, the function is $$f(x)=1-x^3$$.
Now,$$f(3)=1-3^3$$
$$\Rightarrow f(3)=-26$$

Mathematics

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