Determine the intervals where f(x) = -x^2 + 4x is concave up. (2023)

Determine the intervals where f(x) = -x^2 + 4x is concave up. (2023)

Step 1: Start with the function f(x) = -x^2 + 4x.
Step 2: Find the second derivative of the function to determine concavity.
Step 3: First, find the first derivative f'(x) = -2x + 4.
Step 4: Now, find the second derivative f''(x) = -2.
Step 5: Analyze the second derivative. Since f''(x) = -2 is a constant and always negative, it means the function is concave down everywhere.
Step 6: Since the function is never concave up, there are no intervals where f(x) is concave up.

Concavity – Concavity of a function is determined by the sign of its second derivative. If f''(x) > 0, the function is concave up; if f''(x) < 0, it is concave down.
Second Derivative Test – The second derivative test is used to determine the concavity of a function and can indicate points of inflection.