If f'(x) = 4x^3 - 2x and f(1) = 5, find the function f(x).

Answer and explanation

Correct answer: A. f(x) = x^4 - x^2 + 5

  1. Step 1: Integrate f'(x): f(x) = integral (4x^3 - 2x) dx = x^4 - x^2 + C.
  2. Step 2: Use the initial condition f(1) = 5: (1)^4 - (1)^2 + C = 5 implies 0 + C = 5, so C = 5.
Answer
f(x) = x^4 - x^2 + 5.
Why not the others: 'f(x) = x^4 - x^2 + 4' miscalculates C = 5 - 1; 'f(x) = 12x^2 - 2' differentiates instead of integrating; 'f(x) = x^4 - x^2 + 6' adds 1 to 5.

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