1.Relation and Function
normal

Let $f: R \rightarrow R$ 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

(KVPY-2019)

Solution

(d)

Given function $f: R \longrightarrow R$ be a continuous function such that $f\left(x^2\right)=f\left(x^3\right) \forall x \in R$

then $f(x)=f\left(x^{23}\right) \quad$ [on replacing $x$ by $x^{1 / 3}$ ]

Similarly,

$f(x)=f\left(x^{23}\right)=f\left(x^{4 / 9}\right)=f\left(x^{2 / 27}\right)=$

$\quad \ldots=f\left(x^{(23)^n}\right)$

$=f\left(x^0\right) \text { [as } x \text { tends to infinity] }=f(1)$

$\therefore f(x)=f(1)=\text { constant }$

The function $f(x)=$ constant is even and differentiable everywhere.

Standard 12
Mathematics

Similar Questions

Start a Free Trial Now

Confusing about what to choose? Our team will schedule a demo shortly.