​please answer this correctly

​please answer this correctly Question 60 of 90 Iff(x,y) = cos 3xsin4y, findfx(TT/12, 11/6) andfy(TT/12, TT/6). 2. 4. O O o None of the above

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{f}\left(\mathrm{x},\mathrm{y}\right)=\cos 3\mathrm{x} \sin 4\mathrm{y} \\ \mathrm{Hence} \\ {\mathrm{f}}_{\mathrm{x}}\left(\mathrm{x},\mathrm{y}\right)=-3\sin 3\mathrm{x}.\sin 4\mathrm{y} \\ =-3\cos \left(\frac{3\mathrm{\pi }}{12}\right)\sin \left(\frac{4\mathrm{\pi }}{6}\right) \\ =-3\cos \left(\frac{\mathrm{\pi }}{4}\right)\sin \left(\frac{2\mathrm{\pi }}{3}\right) \\ =-3\times \frac{1}{\sqrt{2}}\times \frac{\sqrt{3}}{2} \\ =-\frac{3\sqrt{3}}{2\sqrt{2}}=-\frac{3\sqrt{6}}{4} \\ \mathrm{f}\left(\frac{\mathrm{\pi }}{12},\frac{\mathrm{\pi }}{6}\right)=\cos \left(\frac{3\mathrm{\pi }}{12}\right)\sin \left(\frac{4\mathrm{\pi }}{6}\right) \\ =\cos \left(\frac{\mathrm{\pi }}{4}\right)\sin \left(\frac{2\mathrm{\pi }}{3}\right) \\ =\frac{1}{\sqrt{2}}\times \frac{\sqrt{3}}{2}=\frac{\sqrt{3}}{2\sqrt{2}} \\ {\mathrm{f}}_{\mathrm{y}}\left(\mathrm{x},\mathrm{y}\right)=4\cos 3\mathrm{x}.\cos 4\mathrm{y} \\ =4\times \cos \left(\frac{3\mathrm{\pi }}{12}\right)\cos \left(\frac{4\mathrm{\pi }}{6}\right) \\ =4\cos \left(\frac{\mathrm{\pi }}{4}\right)\cos \left(\frac{2\mathrm{\pi }}{3}\right) \\ =4\times \frac{1}{\sqrt{2}}\times \left(-\frac{1}{2}\right)=-\sqrt{2} \\ \mathrm{Hence} \mathrm{option}\left(4\right) \mathrm{is} \mathrm{correct}. \end{gathered}$$

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