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