For the function f(x) = e^x - x^2, the point of inflection occurs at:
Practice Questions
1 question
Q1
For the function f(x) = e^x - x^2, the point of inflection occurs at:
x = 0
x = 1
x = 2
x = -1
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.
Questions & Step-by-step Solutions
1 item
Q
Q: For the function f(x) = e^x - x^2, the point of inflection occurs at:
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.
Steps: 7
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.
