Question

Let $f:R\to R$ be a continuous function such that $f\left(x^2\right)=f\left(x^3\right)$ 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

The correct option is D both $II$ and $III$ are true

Assuming $x^3=t$ then $x^2=t^{2/3}$ then,
$f\left(t^{2/3}\right)=f\left(t\right)$
Now if we put $t=z^{2/3}$ then
$f\left(z^{4/9}\right)=f\left(z^{2/3}\right)=f\left(z\right)$
Similarly we can write,
$f\left(z\right)=f\left(z^{2/3}\right)=f\left(z^{(2/3{)}^2}\right)=.....=f\left(z^{{\left(2/3\right)}^n}\right)$
So the value of $n$ can move upto infinity so
$n\to \infty ,f\left(z^{{\left(2/3\right)}^n}\right)=f\left(z^0\right)=f\left(1\right)$
Therefore the function is a constant function.
Hence, the given function is an even function and differentiable everywhere.