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Let $f(x)=x^2+x \sin x-\cos x$. Then
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$f(x)=0$ has at least one real root, $f(x)=0$ has at least one positive root, $f(x)=0$ has at least one negative root
$f^{\prime}(x)=2 x+x \cos x+\sin x+\sin x=2(x+\sin x)+x \cos x$
$\Rightarrow f^{\prime}(x) > 0 \quad \forall x > 0, \quad f^{\prime}(x) < 0 \quad \forall x < 0 \Rightarrow f(0)=-1$
$\Rightarrow f^{\prime}(x) > 0 \quad \forall x > 0, \quad f^{\prime}(x) < 0 \quad \forall x < 0 \Rightarrow f(0)=-1$
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