# If # e^x / (5+e^x)#, what are the points of inflection, concavity and critical points?

The point of inflection is

We need

Calculate the first and second derivatives

Therefore,

No critical points.

Therefore,

The sign chart is

Now, calculate the second derivative

We can make the chart

See the graph of the function

graph{e^x/(5+e^x) [-8.89, 8.89, -4.444, 4.445]}

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To find the points of inflection, concavity, and critical points of the function ( \frac{e^x}{5+e^x} ), we first need to find its second derivative and analyze its behavior.

First, let's find the first derivative:

[ f'(x) = \frac{(5+e^x)(e^x) - e^x(e^x)}{(5+e^x)^2} ]

Simplify this to get:

[ f'(x) = \frac{5e^x}{(5+e^x)^2} ]

Now, let's find the second derivative:

[ f''(x) = \frac{d}{dx} \left(\frac{5e^x}{(5+e^x)^2}\right) ]

[ f''(x) = \frac{(5+e^x)^2(5e^x) - 5e^x(2(5+e^x)e^x)}{(5+e^x)^4} ]

[ f''(x) = \frac{5e^x(5+e^x - 10e^x)}{(5+e^x)^3} ]

[ f''(x) = \frac{5e^x(5 - 9e^x)}{(5+e^x)^3} ]

The critical points occur where the first derivative equals zero or is undefined. Setting ( f'(x) = 0 ), we get:

[ \frac{5e^x}{(5+e^x)^2} = 0 ]

Solving this equation yields no real solutions. So, there are no critical points.

The concavity of the function changes where the second derivative equals zero or is undefined. Setting ( f''(x) = 0 ), we get:

[ 5 - 9e^x = 0 ]

Solving this equation yields:

[ e^x = \frac{5}{9} ]

[ x = \ln\left(\frac{5}{9}\right) ]

To determine the concavity at this point, evaluate ( f''(x) ) around ( x = \ln\left(\frac{5}{9}\right) ).

If ( f''(x) > 0 ), the function is concave up. If ( f''(x) < 0 ), the function is concave down.

Substitute ( x = \ln\left(\frac{5}{9}\right) ) into ( f''(x) ) and determine its sign.

Since ( e^x > 0 ) for all real numbers ( x ), ( 5 - 9e^x ) is negative when ( x = \ln\left(\frac{5}{9}\right) ). Therefore, the function is concave down at this point.

As for points of inflection, since the function changes concavity at ( x = \ln\left(\frac{5}{9}\right) ), this point is also a point of inflection.

In summary:

- There are no critical points.
- The function is concave down at ( x = \ln\left(\frac{5}{9}\right) ).
- ( x = \ln\left(\frac{5}{9}\right) ) is a point of inflection.

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