Determine the local maxima of f(x) = -x^2 + 4x. (2022)

Determine the local maxima of f(x) = -x^2 + 4x. (2022)

Step 1: Write down the function you want to analyze: f(x) = -x^2 + 4x.
Step 2: Find the derivative of the function, which tells us the slope of the function. The derivative is f'(x) = -2x + 4.
Step 3: Set the derivative equal to zero to find critical points. So, we solve -2x + 4 = 0.
Step 4: Solve for x. Rearranging gives us -2x = -4, which simplifies to x = 2.
Step 5: Now, we need to find the value of the function at this critical point. Substitute x = 2 into the original function: f(2) = -2^2 + 4(2).
Step 6: Calculate f(2). This gives us f(2) = -4 + 8 = 4.
Step 7: Since the function is a downward-opening parabola (because the coefficient of x^2 is negative), the critical point at x = 2 is a local maximum.

Finding Local Maxima – The process of determining points where a function reaches a local maximum by using the first derivative test.
Critical Points – Identifying points where the derivative is zero or undefined to find potential local maxima or minima.
Second Derivative Test – A method to confirm whether a critical point is a local maximum, minimum, or neither by evaluating the second derivative.