Let `a in R` and let `f: Rvec` be given by `f(x)=x^5-5x+a ,` then (a) `f(x)` has three real roots if `a >4` (b)`f(x)` has only one real roots if `a >4` (c)`f(x)` has three real roots if `a<-4` (d)`f(x)` has three real roots if `-4

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`