Hey! So, I was wondering if someone could answer this. I am mad studying for my exam and I still have a few questions left that I just don't have time to answer, and the assignment is due tomorrow. Thanks and here is the question: Use calculus and algebraic methods to do a complete analysis (i.e., intercepts, critical points, intervals of increase and decrease, points of inflection, intervals of concavity, local maximum or minimum points) for each of the following functions and then sketch the graph of the function. You can use graphing software (Geometer’s Sketchpad or Winplot) to verify your analysis and graph. Keep in mind that your analysis is the most important aspect of this question, not the graph.
ƒ(x) = −x^2 + 4x − 3
g(x) = x^3 − 6x^2 + 9x
h(x) = x^4 − 6x^2
p(x) = x^4 − 4x^3
I will do part of one but really think you should try it yourself first and then ask for help with specific questions.
In general look for what it says to look for.
For example the second one
g(x) = x^3 - 6 x^2 + 9x
first of all what happens to this for large negative x and for large positive x?
for large x magnitude it will look like x^3 so it will go off to negative infinity for large negative x and off to positive infinity for large positive x, sloping ever more steeply (concave)
Now where will it be zero?
set it equal to zero
x^3 - 6 x^2 + 9x = 0 = x(x^2-6x+9)=x(x-3)(x-3)
so zero if x = 0 and double zero (just touches the x axis, does not go through) at x = 3
Now we have a pretty good idea what it looks like already.
It snakes up from -oo on the left, goes through (0,0) headed up to the right, then dips down to the x axis, just touching, before heading up to the right.
Now where is it horizontal between x = 0 and x = 3 ?
take derivative and set to zero
dg/dx = 3 x^2 -12 x + 9
set to zero and factor out 3
0 = x^2 -4x + 3
0 = (x-1)(x-3)
so horizontal at x = 1 (and as we knew at x=3)
What is g at that extreme at x = 1?
g = 1^3 - 6*1^2 + 9*1
=1 - 6 + 9
= 4
That should about do it, you can sketch it pretty neatly
Sure! I can help you with that. Let's analyze each function step-by-step.
1. ƒ(x) = −x^2 + 4x − 3:
a. Intercepts: To find the x-intercepts, set ƒ(x) = 0 and solve the equation.
-x^2 + 4x - 3 = 0
Factor or use the quadratic formula to solve for x.
b. Critical Points: Find the derivative of ƒ(x) and solve for x when the derivative equals zero.
c. Intervals of Increase and Decrease: Analyze the sign of the derivative to determine where the function is increasing or decreasing.
d. Points of Inflection: Find the second derivative of ƒ(x) and solve for x when the second derivative changes sign.
e. Intervals of Concavity: Analyze the sign of the second derivative to determine where the function is concave up or concave down.
f. Local Maximum or Minimum Points: Analyze the behavior of the function at the critical points.
2. g(x) = x^3 − 6x^2 + 9x:
Follow the same steps as in ƒ(x) to analyze g(x).
3. h(x) = x^4 − 6x^2:
Analyze h(x) using the same steps as before.
4. p(x) = x^4 − 4x^3:
Follow the same steps as in the previous questions to analyze p(x).
After completing the analysis for each function, sketch the graphs based on the information obtained. Use a graphing software like Geometer’s Sketchpad or Winplot to help you visualize the graphs and verify your analysis.
Remember, the analysis is the most important part of this question, so make sure you show all the steps and provide clear explanations for each analysis component. Good luck with your exam!
To analyze the given functions and sketch their graphs, we will need to perform several steps. Let's start by analyzing the function ƒ(x) = −x^2 + 4x − 3.
Step 1: Find the intercepts
To find the x-intercepts, set ƒ(x) = 0 and solve for x:
−x^2 + 4x − 3 = 0
This equation can be solved using factoring, completing the square, or the quadratic formula. Once you find the x-intercepts, substitute these values into the function to find the corresponding y-intercepts.
Step 2: Determine critical points
To find the critical points of a function, we need to find where the derivative is equal to zero or undefined. Take the derivative of ƒ(x) and set it to zero:
ƒ'(x) = -2x + 4
-2x + 4 = 0
Solve for x to find the critical point(s).
Step 3: Identify intervals of increase and decrease
To determine where the function is increasing or decreasing, analyze the sign of the derivative in different intervals. Test the derivative in each interval between the critical points found in step 2, and determine whether it is positive or negative.
Step 4: Determine points of inflection
To find the points of inflection, we need to find where the second derivative is equal to zero or undefined. Take the derivative of the derivative of ƒ(x) and set it to zero:
ƒ''(x) = -2
Since -2 is a constant, it is never equal to zero. Therefore, there are no points of inflection in this case.
Step 5: Identify intervals of concavity
To determine the intervals of concavity, analyze the sign of the second derivative in different intervals. Test the second derivative in each interval between the critical points found in step 3 and determine whether it is positive or negative.
Step 6: Find local maximum or minimum points
To find the local maximum or minimum points, examine the behavior of the function at its critical points. You can determine whether the critical point is a local maximum or minimum by considering the intervals of increase and decrease on each side of the critical point.
Once you have completed the analysis, you can sketch the graph of the function using the information obtained. You may also use graphing software like Geometer's Sketchpad or Winplot to verify your analysis and graph.
Repeat these steps for the remaining functions g(x), h(x), and p(x), and sketch their graphs accordingly. Remember that the analysis, including the intercepts, critical points, intervals of increase and decrease, points of inflection, intervals of concavity, and local maximum or minimum points, is the most important aspect of the question.