Let f ( x ) = x 2 + λ x + μ cos x , λ being an integer and μ a real number. The number of ordered pairs ( λ , μ ) for which the equations f ( x ) = 0 and f ( f ( x )) = 0 have the same (non empty) set

Question

Let f ( x ) = x 2 + λ x + μ cos x , λ being an integer and μ a real number. The number of ordered pairs ( λ , μ ) for which the equations f ( x ) = 0 and f ( f ( x )) = 0 have the same (non empty) set of real roots is

TestHub · Accepted Answer

Let α be a root of f ( x ) = 0 , so we have f ( α ) = 0 and thus f ( f ( α )) = 0 , ⇒ f ( 0 ) = 0 ⇒ μ = 0 . We then have f ( x ) = x ( x + λ ) and thus α = 0 , − λ . f ( f ( x )) = x ( x + λ ) ( x 2 + λ x + λ ) We want λ such that x 2 + λ x + λ has no real roots besides 0 and − λ . We can easily find that 0 ≤ λ < 4 .

Mathematics - Quadratic Equation Question with Solution | TestHub

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