In this exercise, we're working with the second derivative $f''(x)$ of a function to determine its concavity and find inflection points. Here’s a breakdown of what’s provided and where the calculations have led so far:

Step-by-Step Explanation

1. Second Derivative Calculation:

   • The second derivative is given as $f''(x) = (5 + 5x)e^x + 5e^x$.
   • This simplifies to $f''(x) = 5e^x (x + 2)$, as factored in the solution.

2. Concavity Analysis:

   • To determine where $f(x)$ is concave up or concave down, we analyze the sign of $f''(x)$:
     • $f''(x) > 0$ when $x > -2$, indicating $f$ is concave upward on the interval $(-2, \infty)$.
     • $f''(x) < 0$ when $x < -2$, indicating $f$ is concave downward on the interval $(-\infty, -2)$.

3. Inflection Point:

   • An inflection point occurs where the concavity changes, which is at $x = -2$.
   • However, the $y$-coordinate for the inflection point is missing in the provided answer. It should be evaluated by plugging $x = -2$ into the original function $f(x)$ to find the complete coordinates of the inflection point.

4. Graph Sketch:

   • The information about concavity is used to sketch the curve, showing a transition from concave down to concave up at $x = -2$, matching the inflection point.

Would you like a detailed explanation on calculating the inflection point’s exact coordinates, or do you have questions on any of these steps?

Related Questions to Explore

1. How do you find the exact value of $y$ for an inflection point?
2. What is the significance of concavity in determining the shape of a graph?
3. How would you interpret the behavior of a function when $f''(x)$ is zero but does not change sign?
4. Why is it necessary to analyze both $f'(x)$ and $f''(x)$ when studying the shape of $f(x)$?
5. How would this approach differ if $f''(x)$ did not factorize cleanly?

Tip

When determining intervals of concavity, remember that the function changes concavity only where the second derivative changes sign, not necessarily wherever it is zero.