Newton Hunts the Point Where the Area Hits 16

A running cost is defined as an accumulated area. For $x \ge 1$,

$A(x) \;=\; \int_{1}^{x} \big(3t^2 - 6t + 4\big)\,dt$

An engineer needs the value of $x$ at which the accumulated cost first reaches exactly $16$ — that is, the root of

$g(x) \;=\; A(x) - 16 \;=\; 0$

She solves it with Newton's method for root finding,

$x_{n+1} \;=\; x_n \;-\; \frac{g(x_n)}{g'(x_n)}$

starting from $x_0 = 5$.

Report $x_2$ — the iterate after exactly two updates. Newton has not converged by then, so $x_2$ is not the exact root, and the exact root is not what is being asked for.

Turn the integral into a closed-form expression for $A(x)$ before you start, evaluate $g$ and $g'$ afresh at each iterate, and keep at least four decimal places in your intermediate work.

Give your answer rounded to two decimal places — for example 1.23.