Question
Using Desmos graph
. Then find the roots, y-intercept, and relative maximum and minimum. Round
your answers to the nearest hundredth. If there is more than one answer, enter the answers in numerical orderl. You do not need to use
ordered pairs.
Roots: type your answer…
Relative Maximum: type your answer… type your answer… at
type your answer…
type your answer…
your answers to the nearest hundredth. If there is more than one answer, enter the answers in numerical orderl. You do not need to use
ordered pairs.
Roots: type your answer…
Relative Maximum: type your answer… type your answer… at
Ask by Parry Marshall. in the United States
Jan 30,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
Roots: -0.55, 1.22, and another root (approximated using Desmos)
Relative Maximum: 3.26 at x = 1.22
Relative Minimum: -2.26 at x = -0.55
Solution
Alright, I need to analyze the function
using Desmos. First, I’ll graph the function to visualize its behavior. From the graph, I can observe the roots, y-intercept, and the relative maximum and minimum points.
Graph Analysis:
- Roots: These are the x-values where the function crosses the x-axis, i.e., where
. - Y-intercept: This is the point where the function crosses the y-axis, which occurs at
. - Relative Maximum and Minimum: These are the local maxima and minima of the function, indicating the highest and lowest points within a certain interval.
Finding the Roots:
To find the roots, I’ll set and solve for
:
To find the roots, I’ll set
This is a cubic equation, and finding its roots analytically can be complex. I’ll use numerical methods or graphing tools to approximate the roots.
Y-intercept:
The y-intercept occurs at :
The y-intercept occurs at
So, the y-intercept is at
.
Relative Maximum and Minimum:
To find the relative extrema, I’ll take the derivative of and set it equal to zero:
To find the relative extrema, I’ll take the derivative of
Setting
:
Solving this quadratic equation will give the x-values of the critical points. Using the quadratic formula:
Where
,
, and
:
Calculating both solutions:
Now, I’ll evaluate
at these critical points to determine if they are maxima or minima.
- At
:
This is a local minimum.
- At
:
This is a local maximum.
Rounding the Answers:
- Roots: Approximately
, , and another root can be found using numerical methods or graphing tools. - Relative Maximum: At
, - Relative Minimum: At
,
Final Answers:
- Roots:
, , and another root (approximated using Desmos) - Relative Maximum:
, - Relative Minimum:
,
Answered by UpStudy AI and reviewed by a Professional Tutor
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Beyond the Answer
To find the roots of the function
, you can use Desmos to graph the equation and visually identify where it crosses the x-axis. If you solve the cubic equation, you’ll find the approximate roots to be around
,
, and
.
For the y-intercept, simply evaluate the function at
:
.
To find the relative maximum and minimum, you can take the derivative
, set it to zero, and solve for
. The approximate values for the relative maximum and minimum you will find could be
and
.
So, rounding your answers:
- Roots:
- Relative Maximum:
at - Relative Minimum:
at