For the function $f\left(x\right)=-x^2+4x+1$, at which $x$-value does a local maximum occur?

• A. $x=0$
• B. $x=-2$
• C. $x=2$
• D. $x=4$

Answer: (C)

Verified step by step guidance

1. Step 1: Identify that the function f(x) = -x^2 + 4x + 1 is a quadratic function, and since the coefficient of x^2 is negative (-1), the parabola opens downward, meaning it has a local maximum at its vertex.
2. Step 2: Recall the formula for the x-coordinate of the vertex of a quadratic function f(x) = ax^2 + bx + c, which is given by x = -b / (2a).
3. Step 3: Substitute the values of a = -1 and b = 4 into the vertex formula: x = -4 / (2 * -1).
4. Step 4: Simplify the expression to find the x-coordinate of the vertex, which represents the x-value where the local maximum occurs.
5. Step 5: Verify that the x-value obtained matches one of the options provided in the problem.