Question

use the graph to estimate the open intervals on which the function is increasing or decreasing. then find the open intervals analytically. (enter your answers using interval notation.)
$y=-(x + 1)^2$
increasing

decreasing

Explanation:

Step1: Differentiate the function

Given \(y = -(x + 1)^2=-x^{2}-2x - 1\).
Using the power rule \((x^n)^\prime=nx^{n - 1}\), we have \(y^\prime=-2x-2=-2(x + 1)\).

Step2: Find the critical points

Set \(y^\prime = 0\), so \(-2(x + 1)=0\).
Solving for \(x\), we get \(x=-1\).

Step3: Determine the sign of \(y^\prime\) in intervals

• For the interval \((-\infty,-1)\), let's take a test - point \(x=-2\). Then \(y^\prime=-2(-2 + 1)=2>0\).
• For the interval \((-1,\infty)\), let's take a test - point \(x = 0\). Then \(y^\prime=-2(0 + 1)=-2<0\).

Answer:

Increasing: \((-\infty,-1)\)
Decreasing: \((-1,\infty)\)