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Let a in R and f : R rarr R be given by...

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`

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`

Text Solution

Verified by Experts

The correct Answer is:
D
`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`.
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