Question

# Find the local maxima and local minima of the function $$f(x)=sin x-cos x, 0<x<2\pi$$. Also find the local maximum and local minimum values.

Solution

## Given $$f(x)=\sin x-\cos x$$differentiate on both sides$$f^{ ' }\left( x \right) =\sin { x } +\cos { x }$$$$f^{ ' }\left( x \right) =0\Rightarrow x=\dfrac { 3\pi }{ 4 } ,\dfrac { 7\pi }{ 4 }$$local maxima:$$f^{ '' }\left( x \right) =\cos { x } -\sin { x }$$$$f^{ '' }\left( \dfrac { 3\pi }{ 4 } \right) =-\sqrt { 2 } <0,$$$$f\left( \dfrac { 3\pi }{ 4 } \right) =\sqrt { 2 }$$local minima:$$f^{ '' }\left( \dfrac { 7\pi }{ 4 } \right) =\sqrt { 2 } >0,\quad f\left( \dfrac { 3\pi }{ 4 } \right) =-\sqrt { 2 }$$$$\sqrt { 2 }$$ and $$-\sqrt { 2 }$$ are the maximum and minimum values.  Mathematics

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