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Find the local maxima and local minima, of the function f(x) = sin x − cos x, 0 < x < 2π. - Mathematics

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Find the local maxima and local minima, of the function f(x) = sin x − cos x, 0 < x < 2π.

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Solution

We have

`f(x)=sinx−cosx             0<x<2π `

`f'(x)=ddx(sinx−cosx)        `

`=cosx+sinx`

For maxima and minima, we have

`f'(x)=0`

`⇒cosx+sinx=0`

`⇒cosx=−sinx`

`⇒x=(3π)/4,(7π)/4`

Now, 

`f"(x)=d/dx(cosx+sinx)                   `

`=−sinx+cosx`

`"At " x=(3π)/4`

`f"((3π)/4)=−sin((3π)/4)+cos((3π)/4)`

`=-1/sqrt2-1/sqrt2`

`=-sqrt2`

`⇒f"((3π)/4)<0`

Thus`x=(3π)/4`  is the point of local maxima.

Local maximum value `f((3π)/4)`

`=sin((3π)/4)−cos((3π)/4)`

`=1/sqrt2+1/sqrt2=sqrt2`

`At  x=(7π)/4`

 

`f"((7π)/4)=−sin((7π)/4)+cos((7π)/4)`

`=1/sqrt2+1/sqrt2=sqrt2`

`⇒f"((7π)/4)>0`

Thus`x=(7π)/4` is the point of local minima.

Local minimum value of `f(x)=f((7π)/4)`

`sin((7π)/4)-cos((7π)/4)`

`=-1/sqrt2-1/sqrt2`

`=-sqrt2`

Concept: Maximum and Minimum Values of a Function in a Closed Interval
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