If $f: \mathrm{R} \rightarrow \mathrm{R}$ is a function defined by $f(\mathrm{x})=[\mathrm{x}-1] \cos \left(\frac{2 \mathrm{x}-1}{2}\right) \pi$, where $[$.$] denotes the greatest integer function, then f$ is...

The number of points, at which of function $f(x)=|2 x+1|-3|x+2|+\left|x^2+x-2\right|, x \in R$ is not differentiable, is $\_\_\_\_$

If $f(x)=\left\{\begin{array}{cc}x+a, & x \leq 0 \\ |x-4|, & x>0\end{array}\right.$ and $g(x)=\left\{\begin{array}{cc}x+1, & x<0 \\ (x-4)^2+b, & x \geq 0\end{array}\right.$ are continuous on $R$, then...

If the function $f(x)=\left\{\frac{\log e\left(1-x+x^2\right)+\log e\left(1+x+x^2\right)}{\sec x-\cos x}, x \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)-\{0\}\right.$ is continuous at $x=0$, then $k$ is equal to: ${ }_{x=0}^k$ :

Let $...

Let $f: R \rightarrow R$ be a continuous function such that $f(3 x)-f(x)=x$. If $f(8)=7$, then $f(14)$ is equal to:

Given $\pi a^2-\pi a b=30 \pi$ and $\pi a b-\pi b^2=18 \pi$ on subtracting, we get $(a-b)^2=a^2-2 a b+b^2=12$ Let $f$ and $g$ be twice...

Let $f, g: R \rightarrow R$ be functions defined by $f(x)=\left\{\begin{array}{cc}{[x],} & x<0 \\ |1-x|, & x \geq 0\end{array}\right.$ and $...

Let $f(x)=\left[2 x^2+1\right]$ and $g(x)=\left\{\begin{array}{ll}2 x-3, & x<0 \\ 2 x+3, & x \geq 0\end{array}\right.$, where $[t]$ is the greatest integer $\leq t$. Then, in...

Let $c, k \in R$. If $f(x)=(c+1) x^2+\left(1-c^2\right) x+2 k$ and $f(x+y)=f(x)+f(y)-x y$, for all $x, y \in R$, then the value of $...