Calculate Area Below y=(x-4)²-2x: Quadratic Function Analysis

To solve this problem, we need to determine where the function $y = (x-4)^2 - 2x$ is decreasing. We'll do this by finding where the first derivative is negative. Here's a detailed step-by-step explanation:

• Step 1: Compute the derivative of the function.

We start with the original function:

$y = (x-4)^2 - 2x$

Expanding the square yields:

$y = (x^2 - 8x + 16) - 2x = x^2 - 10x + 16$

Now, let's find the derivative:

$\frac{dy}{dx} = \frac{d}{dx}(x^2 - 10x + 16) = 2x - 10$

• Step 2: Determine the values where the derivative is negative.

We need to solve the inequality:

$2x - 10 < 0$

Solving for $x$:

$2x < 10$

$x < 5$

Therefore, the function is decreasing for $x < 5$.

Conclusion: The function $y = (x-4)^2 - 2x$ is decreasing for $x < 5$.

The correct choice from the options given is therefore:

$x < 5$