How do you use the second derivative test to find the local extrema for #f(x)=sin(x)#?

Answer 1

Local maxima: #(pi/2+2npi, 1)# Local minima: #((3pi)/2+2npi, -1)#

The function should be differentiated, equated to zero, and solved.

#f(x)=sinx#
#f'(x)=cosx#
#cosx=0 -> x=pi/2+npi# where #n# is an integer.
Now, the second derivative tells us that if #a# is a critical value of #f(x)# (a value which causes the derivative to go to zero), if #f''(a)>0,# there is a minimum at #(a,f(a)),# and if #f''(a)<0,# then there is a maximum at #(a, f(a))#.
As seen above, this function has infinitely many critical values due to the periodic nature of the function. But let's take #x=pi/2, (3pi)/2# and test them in #f''(x).#
#f''(x)=-sinx#
#f''(pi/2)=-sin(pi/2)=-1<0#
So, at #x=pi/2+2npi# there are maxima.
#f''((3pi)/2)=-sin((3pi)/2)=-(-1)=1>0#
So, at #x=(3pi)/2 + 2npi,# there are minima.
#f(pi/2)=1# so the local maxima are #(pi/2+2npi, 1)#
#f((3pi)/2)=-1# so there are local minima at #((3pi)/2+2npi, -1)#
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Answer 2

To use the second derivative test to find the local extrema for ( f(x) = \sin(x) ):

  1. Find the first derivative: ( f'(x) = \cos(x) ).
  2. Find the second derivative: ( f''(x) = -\sin(x) ).
  3. Set ( f''(x) = 0 ) to find critical points: ( -\sin(x) = 0 ) has solutions at ( x = n\pi ), where ( n ) is an integer.
  4. Test these critical points using the second derivative test:
    • If ( f''(x) > 0 ) at a critical point, it's a local minimum.
    • If ( f''(x) < 0 ) at a critical point, it's a local maximum.
    • If ( f''(x) = 0 ), the test is inconclusive.

For ( f(x) = \sin(x) ), the critical points are at ( x = n\pi ).

  • At ( x = 0, ) ( f''(0) = 0 ), so the test is inconclusive at this point.
  • At ( x = \pi, ) ( f''(\pi) = -1 ), so there is a local maximum at ( x = \pi ).
  • At ( x = 2\pi, ) ( f''(2\pi) = 0 ), so the test is inconclusive at this point.

Therefore, the local maximum for ( f(x) = \sin(x) ) occurs at ( x = \pi ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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