The second derivative of a function his given by h''(x)=(sqrt [3](x))/(e^x)+1 Where does the graph of h have a point of inflection? Use a graphing calculator.

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The second derivative of a function his given by h''(x)=(sqrt [3](x))/(e^x)+1 Where does the graph of h have a point of inflection? Use a graphing calculator.

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Answer 1

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 \( x \) where \( h''(x) = 0 \) or where \( h''(x) \) is undefined.

1. Set the second derivative equal to zero:
\[
\frac{\sqrt[3]{x}}{e^x} + 1 = 0
\]
Solving for \( x \):
\[
\frac{\sqrt[3]{x}}{e^x} = -1
\]
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 \( \frac{\sqrt[3]{x}}{e^x} \) 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 \( x = -1 \) and \( x = 1 \). 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 \( x \) where the second derivative \( h''(x) = 0 \), which can be approximated using a graphing calculator.

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The second derivative of a function his given by h''(x)=(sqrt [3](x))/(e^x)+1 Where does the graph of h have a point of inflection? Use a graphing calculator.

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