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 $x\, byx^{1 /3}]$
Similarly,
$f \left(x\right)=f \left(x^{2 /3}\right) =f \left(x^{4 /9}\right)=f \left(x^{8/ 27}\right)= \ldots=f \left(x^{\left(2 /3\right)^n}\right)$
$=f \left(x^{o}\right)$ [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