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Q. The greatest value of $ si{{n}^{3}}x+co{{s}^{3}}x $ is

J & K CETJ & K CET 2007Application of Derivatives

Solution:

Let $ y={{\sin }^{3}}x+{{\cos }^{3}}x $
$ \frac{dy}{dx\,}=3{{\sin }^{2}}\,x\,\cos x-3\,{{\cos }^{2}}x\,\sin x $
$ =3\sin x\,\cos x(\sin x-\cos x) $ Put $ \frac{dy}{dx}=0 $
$ \Rightarrow $ $ 3\,\sin x\,\cos x(sin\,x-cos\,x)=0 $
$ \Rightarrow $ $ x=0,\,\frac{\pi }{2} $ or $ \frac{\pi }{4} $
Now, y has its maximum value at
$ x=0 $ or $ \frac{\pi }{2}, $
and $ {{y}_{\max }}=1 $