Find Decreasing Intervals of y = (3x+3)(9-x): Complete Function Analysis

To solve this problem, we'll determine where the quadratic function is decreasing:

• Step 1: Expand the expression into standard form.
• Step 2: Find the derivative of the function.
• Step 3: Solve the inequality where the derivative is less than zero.
• Step 4: Verify the intervals found.

Let's begin with Step 1:

Expand $y = (3x + 3)(9 - x)$:

$y = 3x(9 - x) + 3(9 - x) = 27x - 3x^2 + 27 - 3x$

$y = -3x^2 + 24x + 27$

Step 2: Differentiate $y$ with respect to $x$:

$\frac{dy}{dx} = -6x + 24$

Step 3: Find where $\frac{dy}{dx} < 0$:

Solving $-6x + 24 < 0$:

$-6x < -24$

$x > 4$

Step 4: Verify:

The function decreases for values of $x$ greater than 4. This matches one of our choices.

Therefore, the interval where the function is decreasing is:

$x > 4$.