For a real number x let [x] denote largest number less than or equal to x . For x ∈ R let f(x)=[x]

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5. Continuity and Differentiation

normal

For a real number $x$ let $[x]$ denote the largest number less than or equal to $x$. For $x \in R$ let $f(x)=[x] \sin \pi x$. Then,

A

$f$ is differentiable on $R$.

B

$f$ is symmetric about the line $x=0$.

C

$\int \limits_{-3}^3 f(x) d x=0$

D

For each real $\alpha$, the equation $f(x)-\alpha=0$ has infinitely many roots.

(KVPY-2014)

Solution

(d)

We have, $f(x)=[x] \sin \pi x$ Graph of $f(x)$ are

Clearly, $f(x)$ is not differentiable at $x=1$ $f(x)$ is not symmetric about line $x=0$

$\int \limits_{-3}^3 f(x) d x \neq 0$

$f(x)=\alpha$ will have infinite solutions.

Standard 12

Mathematics

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