Let the function f: (0, pi) rightarrow R be d | vedclass

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(B) Given $f(	heta) = (\sin 	heta + \cos 	heta)^2 + (\sin 	heta - \cos 	heta)^4$.Using $(\sin 	heta + \cos 	heta)^2 = 1 + \sin 2	heta$ and $(\

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$0.50$

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(B) Given $f(	heta) = (\sin 	heta + \cos 	heta)^2 + (\sin 	heta - \cos 	heta)^4$.Using $(\sin 	heta + \cos 	heta)^2 = 1 + \sin 2	heta$ and $(\sin 	heta - \cos 	heta)^2 = 1 - \sin 2	heta$,we have:$f(	heta) = (1 + \sin 2	heta) + (1 - \sin 2	heta)^2$$f(	heta) = 1 + \sin 2	heta + 1 - 2\sin 2	heta + \sin^2 2	heta = \sin^2 2	heta - \sin 2	heta + 2$.Let $x = \sin 2	heta$. Then $g(x) = x^2 - x + 2$. The derivative $g'(x) = 2x - 1$. Setting $g'(x) = 0$ gives $x = 1/2$.Thus,$\sin 2	heta = 1/2$. For $	heta \in (0, \pi)$,$2	heta \in (0, 2\pi)$.$2	heta = \pi/6$ or $2	heta = 5\pi/6$.$	heta = \pi/12$ or $	heta = 5\pi/12$.These are the points of local minima. Thus $\lambda_1 = 1/12$ and $\lambda_2 = 5/12$.The sum $\lambda_1 + \lambda_2 = 1/12 + 5/12 = 6/12 = 0.50$.

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