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Show that \(\sin (x) \leq x\) for \(x \geq 0\) and \(\sin (x) \geq x\) for \(x \leq 0\). Also show that \(\cos (x) \geq 1-\frac{1}{2} x^{2}\) for all \(x \in \mathbb{R}\).
in Mathematics by Diamond (71,587 points) | 87 views

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Consider \(f(x)=\sin (x)-x\). Then \(f^{\prime}(x)=\cos (x)-1 \leq 0\). This means that \(f(x)\) is weakly decreasing on \(\mathbb{R}\). Since \(f(0)=0\), we have \(f(x) \leq 0\) for all \(x \geq 0\) and \(f(x) \geq 0\) for all \(x \leq 0\). Now consider \(g(x)=\cos (x)-1+\frac{1}{2} x^{2}\). we have \(g^{\prime}(x)=-\sin (x)+x=-f(x)\). Therefore \(g^{\prime}(x) \geq 0\) for \(x \geq 0\) and \(g^{\prime}(x) \leq 0\) for \(x \leq 0\). It follows that \(g\) is weakly increasing for \(x \geq 0\) and \(g\) is weakly decreasing for \(x \leq 0\). Since \(g(0)=0\) we have \(g(x) \geq 0\) for all \(x \in \mathbb{R}\)

by Diamond (71,587 points)

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