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.
Create a free account to view solution
View Solution FreeMore Quadratic Equation Questions
If the equation $ax^{2} + bx + c = 0$, where $a,b,c \in R$ and $a > 0$ has two real roots $\alpha$ and $\beta$ such t...Let $p$ and $q$ be real numbers such that the parabola $y = x^{2} - 2px + q$ has no common point with the x -axis. Let t...The least value of the experssion 2 log10 x - logx (0.01),for x > 1, is...The roots of the equation x2 + 2√3 x + 3 = 0 are-...If α, β, γ are the roots of equation x3 - 2x2 + 3x - 2 = 0 then is euqal to :-...