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Find the maximum value of f(x) = -x^2 + 4x + 1

Worked through step by step. More calculus.

Answer
5\boxed{5}

Substituting x=2x=2 back into the original equation gives the maximum value, and since the parabola opens downward (negative leading coefficient), this vertex is indeed the maximum.

Problem: Find the maximum value of f(x)=−x2+4x+1f(x) = -x^2 + 4x + 1.

Steps

  1. Find the derivative
    f′(x)=−2x+4f'(x) = -2x + 4
  2. Set the derivative equal to zero to find the critical point
    −2x+4=0-2x + 4 = 0
  3. Solve for x
    x=2x = 2
  4. Evaluate the function at the critical point
    f(2)=−(2)2+4(2)+1f(2) = -(2)^2 + 4(2) + 1
  5. Simplify the expression
    f(2)=−4+8+1=5f(2) = -4 + 8 + 1 = 5

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