Question

Let $f:R\to R$ be a periodic function such that $f\left(T+x\right)=1+{\left(\left[1-3f\left(x\right)+3{\left(f\left(x\right)\right)}^2-{\left(f\left(x\right)\right)}^3\right]\right)}^{\frac{1}{3}}$

where $T$ is a fixed positive number, then period of $f\left(x\right)$ is

• A. $T$
• B. $2T$
• C. $3T$
• D. None of these

Answer 1

Answer: (B)

Given : $f\left(T+x\right)=1+{\left(\left[{\left(1-f\left(x\right)\right)}^3\right]\right)}^{\frac{1}{3}}$
$=1+(1-f(x))$
$\Rightarrow f\left(T+x\right)+f\left(x\right)=2$ ... (1)
$\Rightarrow f\left(2T+x\right)+f\left(T+x\right)=2$ ... (2)
Subtracting (1) from (2)
$f\left(2T+x\right)-f\left(x\right)=0$
$f\left(2T+x\right)-f\left(x\right)=0$
Also, $T$ is positive therefore period of $f\left(x\right)=2T$