Questions: f(x)=11/(14+e^(-x)) Determine interval on which concave is up or down Determine the location of any inflection points

Solution Steps

To determine the intervals on which the function \( f(x) = \frac{11}{14 + e^{-x}} \) is concave up or down, and to find the inflection points, we need to follow these steps:

1. Compute the first derivative \( f'(x) \).
2. Compute the second derivative \( f''(x) \).
3. Analyze the sign of \( f''(x) \) to determine concavity:
   • If \( f''(x) > 0 \), the function is concave up.
   • If \( f''(x) < 0 \), the function is concave down.
4. Find the points where \( f''(x) = 0 \) or \( f''(x) \) is undefined to identify potential inflection points.
5. Verify the inflection points by checking the change in concavity around these points.

The first derivative of the function \( f(x) = \frac{11}{14 + e^{-x}} \) is calculated as follows:

\[ f'(x) = \frac{11 e^{-x}}{(14 + e^{-x})^2} \]

The second derivative is computed to analyze the concavity:

\[ f''(x) = -\frac{11 e^{-x}}{(14 + e^{-x})^2} + \frac{22 e^{-2x}}{(14 + e^{-x})^3} \]

To find the inflection points, we set the second derivative equal to zero:

\[ f''(x) = 0 \implies x = -\log(14) \]

This indicates that there is one inflection point at \( x = -\log(14) \).

We analyze the sign of the second derivative to determine the intervals of concavity:

• For \( x < -\log(14) \), \( f''(x) > 0 \) (concave up).
• For \( x > -\log(14) \), \( f''(x) < 0 \) (concave down).

Thus, the intervals of concavity are:

• Concave up: \( (-\infty, -\log(14)) \)
• Concave down: \( (-\log(14), \infty) \)

Final Answer