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.
Similar Questions
If function $f(x) = x(x + 3) e^{-x/2} ;$ satisfies the rolle's theorem in the interval $[-3, 0],$ then find $C$
If function $f(x) = x(x + 3) e^{-x/2} ;$ satisfies the rolle's theorem in the interval $[-3, 0],$ then find $C$
The abscissa of the points of the curve $y = {x^3}$ in the interval $ [-2, 2]$, where the slope of the tangents can be obtained by mean value theorem for the interval $[-2, 2], $ are
The abscissa of the points of the curve $y = {x^3}$ in the interval $ [-2, 2]$, where the slope of the tangents can be obtained by mean value theorem for the interval $[-2, 2], $ are
Consider the function $f (x) = 8x^2 - 7x + 5$ on the interval $[-6, 6]$. The value of $c$ that satisfies the conclusion of the mean value theorem, is
Consider the function $f (x) = 8x^2 - 7x + 5$ on the interval $[-6, 6]$. The value of $c$ that satisfies the conclusion of the mean value theorem, is
Which of the following function can satisfy Rolle's theorem ?
Which of the following function can satisfy Rolle's theorem ?
Let $f (x)$ and $g (x)$ be two continuous functions defined from $R \rightarrow R$, such that $f (x_1) > f (x_2)$ and $g (x_1) < g (x_2), \forall x_1 > x_2$ , then solution set of $f\,\left( {\,g({\alpha ^2} - 2\alpha )\,} \right) >f\,\left( {\,g(3\alpha - 4)\,} \right)$ is
Let $f (x)$ and $g (x)$ be two continuous functions defined from $R \rightarrow R$, such that $f (x_1) > f (x_2)$ and $g (x_1) < g (x_2), \forall x_1 > x_2$ , then solution set of $f\,\left( {\,g({\alpha ^2} - 2\alpha )\,} \right) >f\,\left( {\,g(3\alpha - 4)\,} \right)$ is