The diagram shows the curve \(y = \frac{x}{1 + 3x^4}\), for \(x \geq 0\), and its maximum point \(M\).

(a) Find the \(x\)-coordinate of \(M\), giving your answer correct to 3 decimal places.

(b) Using the substitution \(u = \sqrt{3}x^2\), find by integration the exact area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = 1\).