Define $f:\mathbb{R}\to \mathbb{R}$ by $$f(x)=\{\begin{array}{ll}(1-\cos x)\sin \left(\frac{1}{x}\right),& x\ne 0\\ 0,& x=0\end{array}$$ Then,$$ (a) $f$ is discontinuous. $$ (b) $f$ is continuous but not differentiable. $$ $$ $$ (c) $f$ is differentiable and its derivative is discontinuous. $$ $$ $$$$$$ $$ (d) $f$ is differentiable and its derivative is continuous.

If two real numbers $x$ and $y$ satisfy $(x+5{)}^2+(y-10{)}^2=196$, then the minimum possible value of $x^2+2x+y^2-4y$ is

If the maximum and minimum values of ${\sin }^{6}x+{\cos }^{6}x,$ as $x$ takes all real values, are $a$ and $b$, respectively, then $a-b$ equals

The expression $$\sum _{k=0}^{10}{2}^k\tan \left(2^k\right)$$ equals

Let $f:\mathbb{R}\to [0,\mathrm{\infty })$ be a continuous function such that $$f(x+y)=f(x)f(y)$$ for all $x,y\in \mathbb{R}$. Suppose that $f$ is differentiable at $x=1$ and $${\frac{df(x)}{dx}|}_{x=1}=2$$ Then, the value of $f(1){\log }_{e}f(1)$ is

For $0\le x<2\pi$, the number of solutions of the equation $${\sin }^{2}x+2{\cos }^{2}x+3\sin x\cos x=0$$ is

Let $$\begin{array}{c}p(x)=x^3-3x^2+2x,x\in \mathbb{R},\\ f_0(x)=\{\begin{array}{ll}{\int }_0^{x}p(t)dt,& x\ge 0,\\ -{\int }_x^{0}p(t)dt,& x<0,\end{array}\\ f_1(x)=e^{f_0(x)},f_2(x)=e^{f_1(x)},\dots ,f_n(x)=e^{f_{n-1}(x)}\end{array}$$ How many roots does the equation $\frac{d{f}_n(x)}{dx}=0$ have in the interval $(-\mathrm{\infty },\mathrm{\infty })?$