1. Solve for x 24x^3-26x^2+9x-1=0, where 0<x<1.?

Let f(x) = 24x³ - 26x² + 9x - 1

f(1/2) = 24(1/2)³ - 26(1/2)² + 9(1/2) - 1 = 0
By Factor Theorem, (2x - 1) is a factor of f(x).

24x³ - 26x² + 9x - 1 = 0
24x³ - 12x² - 14x² + 7x + 2x - 1 = 0
(24x³ - 12x²) - (14x² - 7x) + (2x - 1) = 0
12x²(2x - 1) - 7x(2x - 1) + (2x - 1) = 0
(2x - 1)(12x² - 7x + 1) = 0
(2x - 1)(3x - 1)(4x - 1) = 0
x = 1/2 or x = 1/3 or x = 1/4

f(x) = 24x^3-26x^2+9x-1=0
x=0.25 and x=0.5 are two solutions (from the graph)
http://www.wolframalpha.com/input/?i=graph+24x%5E3-26x%5E2%2B9x-1
There is another solution near x=0.3
(use Newton's method with initial value x0=0.3 to find that solution)


f(x) = 24x^3-26x^2+9x-1
f'(x) = 48x^2-52x+9
x0 = 0.300000000
x1 = x0 - f(x0) / f'(x0) = 0.3 - (0.008) / (-0.12) = 0.366666667
x2 = x1 - f(x1) / f'(x1) = 0.36666667 - (-0.01244444) / (-0.38666667) = 0.334482759
x3 = x2 - f(x2) / f'(x2) = 0.33448276 - (-0.00038575) / (-0.33783591) = 0.333340939
x4 = x3 - f(x3) / f'(x3) = 0.33334094 - (-0.00000254) / (-0.33336375) = 0.333333334
x5 = x4 - f(x4) / f'(x4) = 0.33333333 - (0) / (-0.33333333) = 0.333333333

x= 0.333333333
x= 1/3

x=1/4,1/3,1/2

No