# How do you use the second derivative test to find where the function #f(x) = (5e^x)/(5e^x + 6)# is concave up, concave down, and inflection points?

*concave down* on *concave up* on

The second derivative test allows you to determine the intervals on which a function is concave up or concave down by examining the sign of the second derivative around the inflexion points.

Next, calculate the second derivative by using the quotient and chain rules

FInd the critical poin(s) of the function by calculating

This will get you

Take the natual log of both sides of the equation to get

So, the two intervals that you're going to look at are

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To use the second derivative test for the function ( f(x) = \frac{5e^x}{5e^x + 6} ) to find where it is concave up, concave down, and its inflection points, follow these steps:

- Find the first derivative of ( f(x) ) with respect to ( x ) to get ( f'(x) ).
- Find the second derivative of ( f(x) ) by differentiating ( f'(x) ) with respect to ( x ) to get ( f''(x) ).
- Set ( f''(x) = 0 ) and solve for ( x ) to find any points where the concavity may change, these are potential inflection points.
- Determine the concavity of the function by analyzing the sign of ( f''(x) ) in intervals between critical points found in step 3.
- Use the second derivative test:
- If ( f''(x) > 0 ) for a given interval, the function is concave up on that interval.
- If ( f''(x) < 0 ) for a given interval, the function is concave down on that interval.

- Confirm potential inflection points by checking the concavity on either side of them.

These steps will help you identify where the function ( f(x) ) is concave up, concave down, and its inflection points.

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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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