Question

Let $a\in R$ and let $f:R\to R$ be given by$f\left(x\right)=x^5-5x+a$. Then

• A. $f\left(x\right)$ has three real roots if $a>4$
• B. $f\left(x\right)$ has only 1 real root if $a>4$
• C. $f\left(x\right)$ has 3 real roots if $a<-4$
• D. $f\left(x\right)$ has 1 real root if $a<-4$

Answer 1

Answer: (B), (D)

The correct options are
B $f\left(x\right)$ has only 1 real root if $a>4$
D $f\left(x\right)$ has 1 real root if $a<-4$

$f\left(x\right)=x^5-5x+a$

$f^{{}^{\prime }}\left(x\right)=5x^4-5x$

$=5(x^4-1)=5(x-1)(x+1)(x^2+1)$

$\text{when}x=1,-1$

$f\left(x\right)\text{will have slope 0}$

$f\left(1\right)=a-4$, $f(-1)=a+4$

for $-4<a<4$, $f\left(x\right)\text{has 3 real roots}$

for $a<-4,a>4$,$f\left(x\right)\text{has only 1 real root}$