Question

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

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)
$\Rightarrow f\left(2 T + x\right)+f\left(T + x\right)=2$ ... (2)
Subtracting (1) from (2)
$f\left(2 T + x\right)-f\left(x\right)=0$
$f\left(2 T + x\right)-f\left(x\right)=0$
Also, $T$ is positive therefore period of $f\left(x\right)=2T$