Find the points of local maxima and local minima, if any, and local maximum and local minimum values of `f(x)=sinx+1/2cos2x` , where `0

Explanation

Solution: Given `f(x)=sin x+frac{1}{2} cos 2 x` on `(0, frac{pi}{2})`.


Then, `f^{prime}(x)=cos x-sin 2 x=cos x-2 sin x cos x=cos x(1-2 sin x)`.


Now,


`f^{prime}(x)=0`.


`Rightarrow quad cos x(1-2 sin x)=0`


`Rightarrow quad cos x=0 quad ; quad sin x=frac{1}{2}`


`Rightarrow quad x=frac{pi}{6} in(0, frac{pi}{2}) .`


Also, `f^{prime}(x)` is defined for all `x in(0, frac{pi}{2})`.


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