To find the points of inflection for the function \( h(x) \), we need to analyze the second derivative \( h''(x) = \frac{\sqrt[3]{x}}{e^x} + 1 \).A point of inflection occurs where the concavity of the function changes, which happens when the second derivative changes sign. Therefore, we need to find the values of
where \( h''(x) = 0 \) or where \( h''(x) \) is undefined.1. **Set the second derivative equal to zero:**
Solving for
:
This equation is difficult to solve algebraically, so we will use a numerical method or a graphing calculator to approximate the solution.2. **Check where the second derivative is undefined:** The expression
is defined for all real numbers, so there are no points where the second derivative is undefined.3. **Use a graphing calculator:** Input the function \( h''(x) = \frac{\sqrt[3]{x}}{e^x} + 1 \) into a graphing calculator and look for points where the graph crosses the x-axis. These crossings represent the points where \( h''(x) = 0 \).By using a graphing calculator, you can visually identify the points where the second derivative equals zero, indicating potential points of inflection.For example, you might find that the graph crosses the x-axis at approximately
and
. These would be the points where the function \( h(x) \) has a point of inflection.**Conclusion:**The graph of \( h(x) \) has points of inflection at the values of
where the second derivative \( h''(x) = 0 \), which can be approximated using a graphing calculator.