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Find the value of x for which the function f(x) = e^x + x^2 has a minimum. (2020

Practice Questions

Q1
Find the value of x for which the function f(x) = e^x + x^2 has a minimum. (2020)
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Questions & Step-by-Step Solutions

Find the value of x for which the function f(x) = e^x + x^2 has a minimum. (2020)
  • Step 1: Write down 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 = 0 for x.
  • Step 5: Check the value of x that satisfies the equation. In this case, x = 0 is a solution.
  • Step 6: Verify that this point is a minimum by checking the second derivative or using a test.
  • Derivative and Critical Points – Understanding how to find critical points by setting the derivative of a function to zero.
  • Second Derivative Test – Using the second derivative to confirm whether a critical point is a minimum or maximum.
  • Exponential and Polynomial Functions – Knowledge of the behavior of exponential functions compared to polynomial functions.
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