The number of points where the curve $y=x^5-20x^3+50x+2$ crosses the $x$-axis is $............$.

The least value of $a$ for which the equation $\frac{4}{\sin x} + \frac{1}{1 - \sin x} = a$ has at least one solution on the interval $(0, \pi/2)$ is:

$A$ wire of length $36 \text{ m}$ is cut into two pieces. One piece is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum,and the circumference of the circle is $k \text{ m}$,then $\left(\frac{4}{\pi}+1\right) k$ is equal to ..... .

Let $f$ be a differentiable function satisfying $f(x)=1-2x+\int_{0}^{x}e^{(x-t)}f(t)dt, x\in R$ and let $g(x)=\int_{0}^{x}(f(t)+2)^{15}(t-4)^{6}(t+12)^{17}dt, x\in R.$ If $p$ and $q$ are respectively the points of local minima and local maxima of $g$,then the value of $|p+q|$ is equal to . . . . . . .

$A$ rectangle $ABCD$ is inscribed in the region bounded by $y = \sin x$ and the $x-$axis for $x \in [0, \pi]$ (as shown in the figure). The area of the rectangle is maximum when $'\alpha'$ satisfies:

If $x$ is real,then the minimum value of $\frac{x^2-x+1}{x^2+x+1}$ is