The diagram shows the curve $y=x^3\ln \left(x\right)$ and its minimum point $M.$

Find the exact coordinates of $M.$

The diagram shows the curve $y=x^3\ln \left(x\right)$ and its minimum point $M.$

Find the exact area of the shaded region bounded by the curve, the $x$-axis and the line $x=2.$

The diagram shows the curve $y=\frac{x^2}{1+x^3}$ for $x⩾0,$ and its maximum point $M.$ The shaded region $R$ is enclosed by the curve, the $x$-axis and the lines $x=1$ and $x=p.$

Find the exact value of the $x$-coordinate of $M.$

The diagram shows the curve $y=\frac{x^2}{1+x^3}$ for $x⩾0,$ and its maximum point $M.$ The shaded region $R$ is enclosed by the curve, the $x$-axis and the lines $x=1$ and $x=p.$

Calculate the value of $p$ for which the area of $R$ is equal to $1.$ Give your answer correct to $3$ significant figures.