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# Find the local maxima and local minima, of the function f(x) = sin x − cos x, 0 < x < 2π. - CBSE (Commerce) Class 12 - Mathematics

ConceptMaximum and Minimum Values of a Function in a Closed Interval

#### Question

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

#### 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

Thusx=(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

Thusx=(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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#### APPEARS IN

Solution Find the local maxima and local minima, of the function f(x) = sin x − cos x, 0 < x < 2π. Concept: Maximum and Minimum Values of a Function in a Closed Interval.
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