Derivatives and roots - 2

Calculus Level pending

Given the function h ( x ) = 2 x − 1 2 sin ⁡ x + m ln ⁡ x ( x ∈ ( 0 , + ∞ ) , m ∈ R ) h(x)=2x-\dfrac{1}{2}\sin x+m \ln x\ (x \in (0,+\infty), m \in \mathbb R) .

If there exists x 1 , x 2 ∈ ( 0 , + ∞ ) , x 1 ≠ x 2 x_1, x_2 \in (0,+\infty) , x_1 \neq x_2 such that h ( x 1 ) = h ( x 2 ) h(x_1)=h(x_2) , is it always true that x 1 x 2 m 2 < 4 9 \dfrac{x_1 x_2}{m^2} < \dfrac{4}{9} ?

No Yes

Alice Smith by Alice Smith , 20, China

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