# How do you find the local extrema for #F(x)= sin (x + (π/2) )#?

Local extrema are where the first derivative of the expression is equal to zero.

The first derivative of an expression is an inflection point of the expression wherever it is equal to zero. It is a local maxima or minima (extrema) depending on whether the second derivative is positive or negative. IF the second derivative changes sign, then the point is an inflection point and not an extrema.

This can also be determined from the change in sign of the first derivative around the point.

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To find the local extrema of ( F(x) = \sin(x + \frac{\pi}{2}) ), we need to find where the derivative of ( F(x) ) is zero.

First, find the derivative of ( F(x) ):

[ F'(x) = \frac{d}{dx}[\sin(x + \frac{\pi}{2})] ]

Using the chain rule and the derivative of sine function:

[ F'(x) = \cos(x + \frac{\pi}{2}) ]

[ = -\sin(x) ]

Next, set ( F'(x) ) equal to zero and solve for ( x ):

[ -\sin(x) = 0 ]

[ \sin(x) = 0 ]

[ x = n\pi ]

Where ( n ) is any integer.

These are the critical points of ( F(x) ). To determine whether each critical point is a local maximum, local minimum, or neither, we can analyze the behavior of the function around each critical point. The sine function has a maximum or minimum at these critical points, alternating between maximum and minimum as ( n ) changes from even to odd and vice versa.

Therefore, the local extrema occur at ( x = n\pi ), where ( n ) is an integer.

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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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