Determine the local maxima or minima of f(x) = -x^2 + 4x. (2019)
Practice Questions
Q1
Determine the local maxima or minima of f(x) = -x^2 + 4x. (2019)
Maxima at x=2
Minima at x=2
Maxima at x=4
Minima at x=4
Questions & Step-by-Step Solutions
Determine the local maxima or minima of f(x) = -x^2 + 4x. (2019)
Step 1: Write down the function f(x) = -x^2 + 4x.
Step 2: Find the first derivative f'(x) to determine the slope of the function. The first derivative is f'(x) = -2x + 4.
Step 3: Set the first derivative equal to zero to find critical points: -2x + 4 = 0.
Step 4: Solve for x. Rearranging gives -2x = -4, so x = 2.
Step 5: To determine if this critical point is a maximum or minimum, find the second derivative f''(x). The second derivative is f''(x) = -2.
Step 6: Analyze the second derivative. Since f''(x) = -2 is less than 0, this indicates that the function has a local maximum at x = 2.
Step 7: To find the maximum value, substitute x = 2 back into the original function: f(2) = -2^2 + 4(2) = -4 + 8 = 4.
Finding Critical Points – Identifying where the derivative of the function equals zero to find potential local maxima or minima.
Second Derivative Test – Using the second derivative to determine the concavity of the function at critical points to classify them as maxima or minima.
