Analyze the roots of the equation `(x-1)^3+(x-2)^3+(x-4)^3+(x-5)^3=0` by differentiation method.

Answer

one real root

Explanation

Let `f(x) = (x -1)^(3)+(x -2)^(3)+(x -3)^(3)+(x -4)^(3)+(x -5)^(3)`
`rArr f'(x) = 3(x -1)^(2)+3(x -2)^(2)+3(x -3)^(2)+3(x -4)^(2)+3(x -5)^(2)`
Now for `f'(x) = 0 ` we have ` (x -1)^(2)+(x -2)^(2)+(x -3)^(2)+(x -4)^(2)+(x -5)^(2) = 0, ` which has no real roots.
Also coefficient of `x^(3) is 5`, hence, when `x to infty, f(x) to infty`
and when `x to infty, f(x) to infty`
Hence, graph of `y = f(x)` meets the x-axis only once, hence, equation f(x) = 0 has one real root.