Question

use technology to graph ( g(x)=2 x^{3}+8 x^{2}-3 ).
part a. identify the ( x )-intercepts and the points where the local maximums and local minimums occur.
from least to greatest, ( x )-intercepts are
the local maximum is
the local minimums are

Explanation:

Step1: Identify x - intercepts

x - intercepts are the values of \(x\) where \(y = g(x)=0\). Looking at the first set of values \(x\approx - 5.04,x\approx - 0.18,x\approx0.02\) (these are the points where the graph crosses the \(x\) - axis).

Step2: Identify local maximum

A local maximum is a point where the function changes from increasing to decreasing. The point \((-2.67,15.96)\) is a local maximum (since the \(y\) - value is higher than the neighboring points).

Step3: Identify local minima

Local minima are points where the function changes from decreasing to increasing. The points \((2.17,12.59)\) and \((2.67,-15.96)\) are local minima (as the \(y\) - values are lower than the neighboring points in their respective intervals).

Answer:

1. \(x\approx - 5.04,x\approx - 0.18,x\approx0.02\)
2. \((-2.67,15.96)\)
3. \((2.17,12.59),(2.67,-15.96)\)