Let a R and let f : R R be given by f ( x ) = x 5 - 5 x + a . Then [2014]

Question

Let $a \in R$ and let $f : R \to R$ be given by $f (x) = x^{5} - 5 x + a .$ Then [2014]

Smart Achievers · Accepted Answer

(2, 4)

$f (x) = x^{5} - 5 x + a$

$f (x) = 0$ $\Rightarrow x^{5} - 5 x + a = 0$ $\Rightarrow a = 5 x - x^{5} = g (x)$

$\Rightarrow g (x) = 0$ $when$ $x = 0, 5^{1 / 4}, - 5^{1 / 4}$ and $g' (x) = 0 \Rightarrow x = 1, - 1$

Also, $g (- 1) = - 4$ $and$ $g (1) = 4$

Thus, the graph of $g (x)$ will be as shown below.

From the graph, it is clear that if $a \in (- 4, 4),$

then $g (x) = a$ or $f (x) = 0$ has 3 real roots.

If $a > 4$ or $a < - 4$

then $f (x) = 0$ has only one real root.

$∴$ Options (2) and (4) are the correct options.

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