What is the maximum slope of the function f(x) = -x^2 + 4x?

What is the maximum slope of the function f(x) = -x^2 + 4x?

Correct Answer: 2

Step 1: Identify the function we are working with, which is f(x) = -x^2 + 4x.
Step 2: Find the derivative of the function, which tells us the slope. The derivative f'(x) = -2x + 4.
Step 3: To find where the slope is maximum, set the derivative equal to zero: -2x + 4 = 0.
Step 4: Solve for x. Rearranging gives us 2x = 4, so x = 2.
Step 5: Now, find the slope at x = 2 by substituting x back into the derivative: f'(2) = -2(2) + 4.
Step 6: Calculate f'(2) = -4 + 4 = 0. This means the slope is maximum at this point.
Step 7: The maximum slope value is 2, which occurs at x = 2.

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