અહી f: R rightarrow R એ સતત વિધેય છે કે જેથી | vedclass

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Question

(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)=

Vedclass · Accepted Answer

બંને $II$ અને  $III$ એ સત્ય છે.

Vedclass · Answer

(d)

Given function $f: R \longrightarrow R$ be a continuous function such that $f\left(x^2ight)=f\left(x^3ight) \forall x \in R$

then $f(x)=f\left(x^{23}ight) \quad$ [on replacing $x$ by $x^{1 / 3}$ ]

Similarly,

$f(x)=f\left(x^{23}ight)=f\left(x^{4 / 9}ight)=f\left(x^{2 / 27}ight)=$

$\quad \ldots=f\left(x^{(23)^n}ight)$

$=f\left(x^0ight) 	ext { [as } x 	ext { tends to infinity] }=f(1)$

$	herefore f(x)=f(1)=	ext { constant }$

The function $f(x)=$ constant is even and differentiable everywhere.

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