Find the Ascending Area of y=-(2x+6)²: Domain Analysis

To solve this problem, we'll take the following approach step-by-step:

• Step 1: Differentiate the function.
• Step 2: Set the derivative greater than zero and solve.
• Step 3: Determine the correct interval for $x$.

Now, let's work through each step:

Step 1: Differentiate the function
Given $y = -(2x + 6)^2$, apply the chain rule to differentiate:
We first use the substitution $u = 2x + 6$, so $y = -u^2$. The derivative $\frac{dy}{du} = -2u$.
Since $u = 2x + 6$, the derivative $\frac{du}{dx} = 2$. Applying the chain rule, we have:

$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = -2u \cdot 2 = -4(2x + 6) = -8x - 24$

Step 2: Determine where the derivative is positive
A function is increasing when its derivative is positive, so we set the derivative greater than zero:

$-8x - 24 > 0$

Solve for $x$:

$-8x > 24$

$x < -3$

Step 3: Conclusion and multiple-choice validation
The area where the function is increasing corresponds to values of $x$ less than $-3$. According to the choices provided, the correct answer is:

$-3 > x$, which corresponds to choice

Therefore, the solution to the problem is $-3 > x$.