For the function f(x) = e^x - x^2, the point of inflection occurs at:

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What’s inside this PDF?

Question: For the function f(x) = e^x - x^2, the point of inflection occurs at:

Options:

x = 0
x = 1
x = 2
x = -1

Correct Answer: x = 1

Solution:

To find the point of inflection, we compute f\'\'(x) = e^x - 2. Setting f\'\'(x) = 0 gives e^x = 2, leading to x = ln(2). The closest integer is x = 1.

For the function f(x) = e^x - x^2, the point of inflection occurs at:

Practice Questions

For the function f(x) = e^x - x^2, the point of inflection occurs at:

x = 0
x = 1
x = 2
x = -1

Questions & Step-by-Step Solutions

For the function f(x) = e^x - x^2, the point of inflection occurs at:

Step 1: Start with the function f(x) = e^x - x^2.
Step 2: Find the first derivative f'(x) to analyze the function's behavior.
Step 3: Compute the second derivative f''(x) to find the points of inflection.
Step 4: Set the second derivative f''(x) equal to zero: f''(x) = e^x - 2.
Step 5: Solve the equation e^x - 2 = 0, which simplifies to e^x = 2.
Step 6: Take the natural logarithm of both sides to find x: x = ln(2).
Step 7: Identify the closest integer to ln(2), which is approximately 0.693, so the closest integer is x = 1.

Second Derivative Test – Understanding how to find points of inflection by analyzing the second derivative of a function.
Exponential Functions – Knowledge of properties of the exponential function and its behavior.
Logarithmic Functions – Understanding how to solve equations involving logarithms, particularly in the context of finding x.

For the function f(x) = e^x - x^2, the point of inflection occurs at:

What’s inside this PDF?

For the function f(x) = e^x - x^2, the point of inflection occurs at:

Practice Questions

Questions & Step-by-Step Solutions

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