Question

Identify the open intervals on which the function is increasing or decreasing. $$ y=-(x+1)^{2} $$

Short answer

The function \(y = -(x+1)^{2}\) is decreasing on the entire real line.

Step-by-step solution

Find the derivative of the function

The derivative of \(y = -(x + 1)^{2}\) can be found using the chain rule for differentiation. Therefore, the derivative, \(y'\), is \(y' = -2 * (x + 1)\).

Find the critical points

Critical points occur where the derivative of the function is equal to zero or does not exist. In this case, we set \(y' = 0\) and solve for \(x\). The solution is \(x = -1\). So, we have one critical point.

Test the intervals around the critical points

We divide the real line into intervals based on the critical points, so the intervals we test are \((-∞, -1)\) and \((-1, ∞)\). Choose any number from each interval and substitute into the \(y'\). If \(y' > 0\), then the function is increasing on that interval. If \(y' < 0\), then the function is decreasing on that interval. After the substitution, we will find that the function is decreasing on the entire real line as \(y'\) is always negative.