Let `-1 le ple plt1.` Show that the equation `4x^(3)-3x-p=0` has a unique root in the interval `[1//2.1]` and identify it.

Answer

`[- (2)/(a), (a)/(3)]`

Explanation

Given, ` - 1 le p le 1 `
Let ` f(x) = 4x^(3) - 3x - p =0`
Now, `f(1//2) = (1)/(2) - (3)/(2) - p = -1 - p le 0 " " [ because p ge -1 ]`
Also, ` f(1) = 4 -3 - p = 1-p ge 0 " " [ because p le 1]`
`therefore f(x)` has atleast one real root between `[ 1//2, 1]`
Also, `f ' (x) = 12 x ^(2 ) - 3 gt 0 " on " [1//2, 1]`.
`rArr f' (x) ` increasing on `[ 1//2, 1]`
`rArr f` has only one real root between `[1//2, 1]`.
To find a root, we observe `f (x)` contains `4x ^(3) - 3x,` which is multiple angle formula for `cos 3 theta`.
`therefore ` Put `" " x = cos theta `
`rArr 4 cos ^(3) theta - 3 cos theta - p =0`
`rArr p = cos 3 theta rArr theta = (1//3) cos ^(-1) (p)`
`therefore `Root is ` cos (1/(3) cos ^(-1) (p))`.