Question

Let the function $f :(0, \pi) \rightarrow R$ 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 $\theta \in\left\{\lambda_{1} \pi, \ldots, \lambda_{r} \pi\right\}$, where $0 < \lambda_{1} < \ldots < \lambda_{r} < 1$. Then the value of $\lambda_{1}+\ldots+\lambda_{r}$ is ______

Answer 1

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

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