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,
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,
- [KVPY 2014]
- 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.
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Verify Mean Value Theorem, if $f(x)=x^{3}-5 x^{2}-3 x$ in the interval $[a, b],$ where $a=1$ and $b=3 .$ Find all $c \in(1,3)$ for which $f^{\prime}(c)=0$
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- [JEE MAIN 2021]
The function $f(x) = {x^3} - 6{x^2} + ax + b$ satisfy the conditions of Rolle's theorem in $[1, 3]. $ The values of $a $ and $ b $ are
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Examine if Rolle's Theorem is applicable to any of the following functions. Can you say some thing about the converse of Roller's Theorem from these examples?
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$f(x)=[x]$ for $x \in[-2,2]$