A structural beam's vertical deflection D(x)D(x)D(x) (in mm) at a distance xxx (in metres) from the left support is modeled by the function
D(x)=x4−6x3+5x+12,x≥0 D(x) = x^4 - 6x^3 + 5x + 12, \quad x \ge 0 D(x)=x4−6x3+5x+12,x≥0Show that the equation D(x)=0D(x) = 0D(x)=0 has a root, α\alphaα, in the interval [5.5,6.0][5.5, 6.0][5.5,6.0].
Show that the equation D(x)=0D(x) = 0D(x)=0 can be rewritten in the form
x=6x2−5−12x3 x = \sqrt[3]{6x^2 - 5 - \frac{12}{x}} x=36x2−5−x12The iterative formula
xn+1=6xn2−5−12xn3 x_{n+1} = \sqrt[3]{6x_n^2 - 5 - \frac{12}{x_n}} xn+1=36xn2−5−xn12is used to find α\alphaα.
Starting with x1=5.7x_1 = 5.7x1=5.7 and using the iterative formula,
(i) find, to 4 decimal places, the value of x2x_2x2.
(ii) find, to 4 decimal places, the value of α\alphaα.