At what value of $x$ does the function $f(x) = \sin x(1 + \cos x)$ have a maximum value?

Let $f(x) = x \sin \pi x$,$x > 0$. Then for all natural numbers $n$,$f^{\prime}(x)$ vanishes at
$(A)$ a unique point in the interval $\left(n, n+\frac{1}{2}\right)$
$(B)$ a unique point in the interval $\left(n+\frac{1}{2}, n+1\right)$
$(C)$ a unique point in the interval $(n, n+1)$
$(D)$ two points in the interval $(n, n+1)$

The maximum slope of the curve $y = \frac{1}{2} x^{4} - 5 x^{3} + 18 x^{2} - 19 x$ occurs at the point

$A$ wire of length $28 \, m$ is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?

The absolute minimum value of the function $f(x) = |x^2 - x + 1| + [x^2 - x + 1]$,where $[t]$ denotes the greatest integer function,in the interval $[-1, 2]$,is:

The point in the interval $[0, 2\pi]$, where $f(x) = e^x \sin x$ has maximum slope, is