Let a `in R` and let f: `R to R` be given by f(x) `=x^(5) -5x+a.` then

Options

A. f(x) has three real roots if `agt4`
B. f(X) has only one real root if `agt4`
C. f(x) has three real roots if `alt-4`
D. f(X) has threee real roots if `-4ltalt4`

Answer

2,4

Explanation

Let y = f(X) =`x^(5)-5x`
`rarr F(x)=5x^(4)-5`
`=5(x-1)(x+1)(x^(2)+1)`
`f(X)=20x^(3)`
f(1)=20 and f(-1)=-20
`therefore` x=1 is point of minma and x =-1 is point of maxima
Also f(1) =-4 and f(-1) =4
Graph of y=f(x) is as shown in following figure

Form the graph `x^(5)-5x=-a` has on real root if -`a lt -4 or -a gt4`
i.e `agt4 or alt-4`
`x^(5)-5x=-a` has three real roots if -`4 lt-alt4`
`x^(5)-5x=-a` has three real roots if -`4 lt-alt4`
i.e `-4ltalt4`