Question

Let $f:R\to R$ be a continuous function such that $f(x^2)=f(x^3)$ for all $x\in R$ Consider the following statements.
I. $f$ is an odd function
II. $f$ is an even function
III. $f$ is differentiable everywhere
Then,

• A. I is true and III is false
• B. II is true and III is false
• C. Both I and III are true
• D. Both II and III are true

Answer 1

Answer: (D)

Given function $f:R\to R$ be a continuous function such that
$f\left(x^2\right)=f\left(x^3\right)\forall x\in R$
then $f\left(x\right)=f\left(x^{2/3}\right)$ [on replacing $xby{x}^{1/3}]$
Similarly,
$f\left(x\right)=f\left(x^{2/3}\right)=f\left(x^{4/9}\right)=f\left(x^{8/27}\right)=\dots =f\left(x^{{\left(2/3\right)}^n}\right)$
$=f\left(x^o\right)$ [as $x$ tends to infinity] $=f\left(1\right)$
$\therefore f\left(x\right)=f\left(1\right)$ =constant
The function $f\left(x\right)$ = constant is even and differentiable everywhere