Let a `in` R and f : `R rarr R` be given by `f(x)=x^(5)-5x+a`, then
(a) `f(x)=0` has three real roots if `a gt 4`
(b) `f(x)=0` has only one real root if `a gt 4`
(c) `f(x)=0` has three real roots if `a lt -4`
(d) `f(x)=0` has three real roots if `-4 lt a lt 4`

Options

A. (a) `f(x)` has three real roots if `agt4`
B. (b) `f(x)` has only one real root if `agt4`
C. (c) `f(x)` has three real roots if `alt-4`
D. (d) `f(x)` has three real roots if `-4ltalt4`

Answer

D

Explanation

`f(x)=x^(5)-5x` and `g(x)=-a`
`:.f'(x)=5x^(4)-5`

`=5(x^(2)+1)(x-1)(x+1)`
Clearly `f(x)=g()` has one real root, if `a gt4` and three real roots if `|a|lt4`.