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If f(x) = e^x - x^2, find the x-coordinate of the local maximum.

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Question: If f(x) = e^x - x^2, find the x-coordinate of the local maximum.

Options:

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  2. 1
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  4. 3

Correct Answer: 1

Solution: 

Find f\'(x) = e^x - 2x. Setting f\'(x) = 0 gives a local maximum at x = 1.

If f(x) = e^x - x^2, find the x-coordinate of the local maximum.

Practice Questions

Q1
If f(x) = e^x - x^2, find the x-coordinate of the local maximum.
  1. 0
  2. 1
  3. 2
  4. 3

Questions & Step-by-Step Solutions

If f(x) = e^x - x^2, find the x-coordinate of the local maximum.
  • Step 1: Start with the function f(x) = e^x - x^2.
  • Step 2: Find the derivative of the function, which is f'(x) = e^x - 2x.
  • Step 3: Set the derivative equal to zero to find critical points: e^x - 2x = 0.
  • Step 4: Solve the equation e^x = 2x. This may require numerical methods or graphing.
  • Step 5: Check the critical points to determine if they are local maxima or minima.
  • Step 6: After checking, you find that x = 1 is a local maximum.
  • Differentiation – The process of finding the derivative of a function to determine critical points.
  • Critical Points – Points where the derivative is zero or undefined, indicating potential local maxima or minima.
  • Second Derivative Test – A method to determine the nature of critical points by evaluating the second derivative.
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