dinsdag 2 mei 2017

De gulden snede

\( \begin{array}{l} f(x) = \sin (x)(\sin (x) + 2\cos (x)) \\ f(x) = \sin ^2 (x) + 2\sin (x)\cos (x) \\ f'(x) = 2\sin (x)\cos (x) + 4\cos ^2 (x) - 2 \\ \left. \begin{array}{l} f'(x) = 4\cos ^2 (x) + 2\sin (x)\cos (x) - 2 \\ f'(x) = 0 \\ \end{array} \right\} \Rightarrow \\ 4\cos ^2 (x) + 2\sin (x)\cos (x) - 2 = 0 \\ 2\cos (2x) + 2 + \sin (2x) - 2 = 0 \\ 2\cos (2x) + \sin (2x) = 0 \\ 2 + \tan (2x) = 0 \\ \tan (2x) = - 2 \\ x = \frac{3}{8}\pi - \frac{1}{2}\arctan \left( {\frac{1}{3}} \right) \\ f\left( {\frac{3}{8}\pi - \frac{1}{2}\arctan \left( {\frac{1}{3}} \right)} \right) = \frac{1}{2} + \frac{1}{2}\sqrt 5 = \varphi \\ \end{array} \)

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