Question

use your graphing calculator to identify the points where the local maximums and local minimums. determine the intervals for which the function is increasing or decreasing. (2 points, 1/2 point each)

1. $g(x) = 5x^3 - 4x^2 - 2x + 1$

local maximum(s):
$x=-0.185489$ $y=1.2014435$
local minimum(s):
$x=0.7188207$ $y=-0.647369$
increasing:
decreasing:
solve the equation. (2 points)

Explanation:

Step1: Confirm critical x-values

The critical points (where the function's derivative is zero) are $x \approx -0.185489$ (local max) and $x \approx 0.7188207$ (local min).

Step2: Test intervals for increase/decrease

For $x < -0.185489$, the function's slope is positive (increasing). For $-0.185489 < x < 0.7188207$, the slope is negative (decreasing). For $x > 0.7188207$, the slope is positive (increasing).

Answer:

local maximum(s): $(-0.185489, 1.2014135)$
local minimum(s): $(0.7188207, -0.647369)$
increasing: $(-\infty, -0.185489) \cup (0.7188207, \infty)$
decreasing: $(-0.185489, 0.7188207)$