Question

Assertion :A function $f$ satisfies the condition $f(x+T)=1+{\{1-3f(x)+3(f\left(x\right){)}^2-\left(f\right(x\left){)}^3\right\}}^{1/3}$(where $T$ is a fixed positive number) is periodic with period $2T$. Reason: If $f(x+2T)=f\left(x\right)$ then period of $f\left(x\right)$ is $2T$.

• 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. Both Assertion and Reason are incorrect

Answer 1

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

Given
$f(x+T)=1+{\{1-3f(x)+3{\left(f\left(x\right)\right)}^2-{\left(f\left(x\right)\right)}^3\}}^{1/3}$

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

$f(x+T)=-f\left(x\right)+2$

$\Rightarrow f(x+T)+f\left(x\right)=2$ ....(i)

Substituting $x\to x+T$

$\therefore f(x+2T)+f(x+T)=2$ ......(ii)

$\therefore$ Subtracting (i) and (ii), we get

$f(x+2T)-f\left(x\right)=0$

$\Rightarrow f(x+2T)=f\left(x\right)$

$\Rightarrow$ Period of $f\left(x\right)$ is $2T$

Assertion $\left(A\right)$ & Reason $\left(R\right)$ both are true & Reason $\left(R\right)$ is correct explanation of Assertion $\left(A\right)$.