Ajay
Last Activity: 8 Years ago
Firstly the question is wrong , the interval in which root will be unique is [1/2, 1] instead of [-1/2, 1].
The equation can be written as 4x3-3x = p
Draw the graph of of function y = f(x) = 4x3-3x. by finding its maxima and minima.
The given equation will have unique root in any interval if line y = p cuts the graph y=f(x) at exactly one point.
From the graph it would be clear that that this will be true if interval of x is [1/2, 1]
Let the root be a, so it will satisfy the given equation we have
4a3-3a = p
let a = cosr
4cos3r-3cosr = p
cos3r = p
r = 1/3cos-1p
a = cosr = cos(1/3cos-1p) will be the required root
