Find the maximum and minimum values of the function $f\left(x\right)=x+\sin 2x$ on $[0,2\pi ]$

Find the equation of the tangent to the curve $y=f\left(x\right)$ at $\left(x_0,y_0\right)$, if $\frac{dy}{dx}$ does not exist at this point.

If $f\left(x\right)=x^3-6x^2-36x+7$, find the interval for which $f\left(x\right)$ is strictly increasing.

If $f\left(x\right)=x^3-6x^2-36x+7$, find the interval for which $f\left(x\right)$ is strictly decreasing.

Find the area of the largest rectangle that can be formed having a perimeter of $40$ meters.