Question

Assertion :Let $f\left(x\right)$ be a polynomial function satisfying $f\left(x\right).f\left(\frac{1}{x}\right)=f\left(x\right)+f\left(\frac{1}{x}\right)$. If $f\left(4\right)=65$ and $l_1,l_2,l_3$ are in G.P., then $f^{\prime }\left(l_1\right),f^{\prime }\left(l_2\right),f^{\prime }\left(l_3\right)$, are also in G.P. Reason: $f\left(x\right)=\pm x^n+1$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$f\left(x\right)+f\left(\frac{1}{x}\right)=f\left(x\right)\cdot f\left(\frac{1}{x}\right)\left(1\right)$

We can write $\left(1\right)$ as

$f\left(x\right)=\frac{f\left(\frac{1}{x}\right)}{f\left(\frac{1}{x}\right)-1}\left(2\right)$

We can also write $\left(1\right)$ as

$f\left(\frac{1}{x}\right)=\frac{f\left(x\right)}{f\left(x\right)-1}\left(3\right)$

Multiplying $\left(1\right)$ and $\left(2\right)$ we get

$f\left(x\right)f\left(\frac{1}{x}\right)=\frac{f\left(\frac{1}{x}\right)\cdot f\left(x\right)}{[f\left(\frac{1}{x}\right)-1]\left[f\right(x)-1]}\Rightarrow \left(f\right(x)-1)(f\left(\frac{1}{x}\right)-1)=1\left(4\right)$

Take $g\left(x\right)=f\left(x\right)-1g\left(\frac{1}{x}\right)=f\left(\frac{1}{x}\right)-1$

We can write $\left(4\right)$ as

$g\left(x\right)\cdot g\left(\frac{1}{x}\right)=1$

Now, if $g\left(x\right)$ is a polynomial, $g\left(x\right)=\pm x^n$

$\Rightarrow f\left(x\right)-1=\pm x^n\Rightarrow f\left(x\right)=\pm x^n+1$

Hence, reason is correct

Now, $f\left(4\right)=65\Rightarrow f\left(x\right)=x^3+1f^{\prime }\left(x\right)=3x^2$

Take $l_1=a,l_2=ar,l_3=a{r}^2$

$\Rightarrow$$f^{\prime }\left(l_1\right)=3a^2{f}^{\prime }\left(l_2\right)=3a^2{r}^2{f}^{\prime }\left(l_3\right)=3a^2{r}^4$

$\Rightarrow f^{\prime }\left(l_1\right),f^{\prime }\left(l_2\right),f^{\prime }\left(l_3\right)$ is also in GP

Hence, assertion and reason both are correct, and reason is the correct explanation for assertion.