Determine the point where the function f(x) = 4x - x^2 has a maximum. (2022)

Determine the point where the function f(x) = 4x - x^2 has a maximum. (2022)

Step 1: Write down the function f(x) = 4x - x^2.
Step 2: Find the derivative of the function, which is f'(x) = 4 - 2x.
Step 3: Set the derivative equal to zero to find critical points: 4 - 2x = 0.
Step 4: Solve for x: 2x = 4, so x = 2.
Step 5: To find the maximum value, substitute x = 2 back into the original function: f(2) = 4(2) - (2^2).
Step 6: Calculate f(2): f(2) = 8 - 4 = 4.
Step 7: The maximum point of the function is at (2, 4).

Finding Maximum of a Quadratic Function – The question tests the ability to find the maximum point of a quadratic function using calculus, specifically by finding the derivative and setting it to zero.
Critical Points – Understanding that critical points occur where the derivative is zero or undefined, which is essential for determining maxima and minima.
Evaluating Functions – The question also tests the ability to evaluate the function at the critical point to find the maximum value.