Question

Let the function $f:(0,\pi )\to R$ be defined by
$f(\theta )=(\sin \theta +\cos \theta {)}^2+(\sin \theta -\cos \theta {)}^4\text{. }$
Suppose the function f has a local minimum at $\theta$ precisely when $\theta \in \left\{{\lambda }_1\pi ,\dots ,{\lambda }_r\pi \right\}$, where $0<{\lambda }_1<\dots <{\lambda }_r<1$. Then the value of ${\lambda }_1+\dots +{\lambda }_r$ is ______

Answer 1

Answer: 0.5

$f(\theta )=(\sin \theta +\cos \theta {)}^2+(\sin \theta -\cos \theta {)}^4$
$f^{\prime }(\theta )=2(\sin \theta +\cos \theta )(\cos \theta -\sin \theta )+4(\sin \theta -\cos \theta {)}^3(\cos \theta +\sin \theta )$
$=2(\sin \theta +\cos \theta )(\sin \theta -\cos \theta )(1-2\sin 2\theta )$
$f^{\prime }(\theta )=0\theta =\frac{\pi }{12},\frac{\pi }{4},\frac{5\pi }{12},\frac{3\pi }{4}$

Pts of minima at $x=\frac{\pi }{12},\frac{5\pi }{12}$
${\lambda }_1+{\lambda }_2=\frac{1}{12}+\frac{5}{12}=\frac{1}{2}=0.5$