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. f(x) has three real roots if `a gt4`
B. f(x) has only one real root if `a gt 4`
C. f(x) has three real roots if `a lt-4`
D. f(x) has three real roots if `-4 lt a lt 4`

Answer

B, D

Explanation

Let g(x)`x^(5)-5x` and h(x)=-a. Then, the number of real roots of f(x)=0 is the number of points of intersection of the curves y=g(x) and y=h(x). Clearly, these two curves intersect at three points, if `-4 lt a lt 4`.

So, f(x) has three real roots, if `4 lt a lt 4`.
The two curves y=g(x) and y=h(x) intersect at exactly one point if `a gt 4 " or "alt-4`.
So, f(x) has only one real root if `a lt -4 " or " a gt 4`.