The number of solutions of the equation $16\left({\sin }^{5}x+{\cos }^{5}x\right)=11\left(\sin x+\cos x\right)$ in the interval  $\left[0,2\pi \right]$  is

 $16\left({\sin }^{5}x+{\cos }^{5}x\right)-11\left(\sin x+\cos x\right)=0$

$\Rightarrow \left(\sin x+\cos x\right)\left\{16\left({\sin }^{4}x-{\sin }^{3}x\cos x+{\sin }^{2}x{\cos }^{2}x-\sin x{\cos }^{3}x+{\cos }^{4}x\right)-11\right\}=0$

$\begin{array}{l}\Rightarrow \left(\sin x+\cos x\right)\left\{16\left(1-{\sin }^{2}x{\cos }^{2}x-\sin x\cos x\right)-11\right\}=0\\ \Rightarrow \left(\sin x+\cos x\right)\left(4\sin x\cos x-1\right)\left(4\sin x\cos x+5\right)=0\\ As4\sin x\cos x+5\ne 0,Wehave\\ Therequiredvaluesare\\ \pi /12,5\pi /12,9\pi /12,13\pi /12,17\pi /12,21\pi /12\\ Theyare6solutionson\left[0,2\pi \right]\end{array}$