Let `f(x)=ax^(5)+bx^(4)+cx^(3)+dx^(2)+ ex`, where a,b,c,d,e in R and `f(x)=0` has a positive root. `alpha`. Then,

Options

A. f'(x)=0 has a root `alpha_(1)` such that `0le alpha_(1) le alpha_(0)`
B. f'(x)=0 has at leat one real root
C. f'(x)=0 has at least two real roots
D. all of the above

Answer

D

Explanation

It is given that `alpha` is positive root of f(x) and by inspection, we hae f(0)=0. Therefore, `= 0 and x= alpha` are the roots of f(x)=0. By Rolle's theorem, f'(x)0 has a root `alpha_(1)` between 0 and `alpha i.e., 0 lt alpha_(1) le alpha`.
So ophtion (a) is correct
Clearly, `f'(x)=0` is a fourth degree equation in x and imaginary roots always occure in pairs. Since, `x=alpha_(1)` is a root of f'(x)-0
Clearly, f'(x)=0 will have another real root, `alpha_(2)`(say).
Now, `alpha_(1) and alpha_(2)` real root of f(x)=0. Therefore, by Rollle's therem `f''(x)=0` will have a real root betwen `alpha_(1) and alpha_(2)`
Thus, option (b) is correct
We have seent that `x=0,x= alpha` are two real roots of f(x)0. As f(x)=0 is a fifth degree equation, it will have at leat three realn roots. Consquently, by Rolle's theorem f'(x)=0 will have at least two real roots
Thus, option (c) is also correct.