JEE Advanced | 2014Quadratic EquationHard

Question

Let a ∈ R and let f : R → R be given by f(x) = x5 - 5x + a. Then

Options

A.f(x) has three real roots if a > 4
B.f(x) has only one real root if a > 4
C.f(x) has three real roots if a < - 4
D.f(x) has three real roots if -4 < a < 4

Solution

         
f(x) = x5 - 5x + a
if f(x) = 0     ⇒ a = 5x - x5 = g(x)
       g(x) = 0     ⇒ x = 0, 51/4, - 51/4
       g′(x) = 0     ⇒ 5 - 5x4 = 0 ⇒ x = 1, -1
       g(-1) = -4
       g(1) = 4
If a ∈ (-4, 4) ⇒ f(x) = 0 has 3 real roots
if a > 4 or a < - 4 ⇒ f(x) = 0 has only 1 real root.

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