MonotonicityHard

Question

Number of roots of the equation x2 .e2-IXI = 1 is -

Options

A.2
B.4
C.6
D.infinite

Solution

     
x2 e2- x - 1 = 0
f(x) = - 1 ∀ x ≥ 0
f′(x) =
⇒ f increases in (0, 2), f decreases in (2, ∞)
Also f (0) < 0 & f(2) > 0 ⇒ Exactly one root in (0, 2)
f(x) < 0
⇒ exactly one root in (2, ∞)
⇒ exactly 2 roots in (0, ∞)
⇒ equation has exactly 4 roots
∵ f(x) is even function.

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