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

Local maxima:

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

By signing up, you agree to our Terms of Service and Privacy Policy

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

- Find the first derivative: ( f'(x) = \cos(x) ).
- Find the second derivative: ( f''(x) = -\sin(x) ).
- Set ( f''(x) = 0 ) to find critical points: ( -\sin(x) = 0 ) has solutions at ( x = n\pi ), where ( n ) is an integer.
- 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 ).

By signing up, you agree to our Terms of Service and Privacy Policy

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.

- What are the points of inflection, if any, of #f(x)= x^5+ x^3- x^2-2 #?
- How do you find the inflection point of the function #f(x)=xe^(-2x)#?
- How do you find the local maximum and minimum values of # f ' (x) = (x-8)^9 (x-4)^7 (x+3)^7#?
- What are the points of inflection of #f(x)= 2sin (x) - cos^2 (x)# on #x in [0, 2pi] #?
- How do you find the smallest and largest points of inflection for #f(x)=1/(5x^2+3)#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7