The values of $x$ at the stationary points of $f(x)=x^3+3x^2-2$ are

If $f(x) = (\sin^4 x + \cos^4 x)$,$0 < x < \frac{\pi}{2}$,then the function has a minimum value of $ . . . . . . $ at $x = . . . . . . $.

If $x + y = 10$,then the maximum value of $xy$ is

Find the local maximum and local minimum values for the function $f(x) = \sin x - \cos x$ where $0 < x < 2\pi$.

If the lengths of three sides of a trapezium other than the base are equal to $10 \ cm$,find the maximum area of the trapezium.

Let $R$ denote the set of all real numbers. Let $f: R \rightarrow R$ be defined by $f(x)=\begin{cases} \frac{6x+\sin x}{2x+\sin x} & \text{if } x \neq 0 \\ \frac{7}{3} & \text{if } x=0 \end{cases}$. Then which of the following statements is (are) True?
$(A)$ The point $x=0$ is a point of local maxima of $f$
$(B)$ The point $x=0$ is a point of local minima of $f$
$(C)$ Number of points of local maxima of $f$ in the interval $[\pi, 6\pi]$ is $3$
$(D)$ Number of points of local minima of $f$ in the interval $[2\pi, 4\pi]$ is $1$