use the graph to estimate the open intervals on which the function is increasing or decreasing. then find…

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.)\n$y=-(x + 1)^2$\nincreasing\n\ndecreasing\n

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.)\n$y=-(x + 1)^2$\nincreasing\n\ndecreasing\n

Answer

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))