Help with a calcululs analysis problem

[PatrickrG]

Hello, I need to do a complete analysis using calculus and algebraic methods for these problems

1. ƒ(x) = −x2 + 4x − 3
2. g(x) = x3 − 6x2 + 9x
3. h(x) = x4 − 6x2
4. p(x) = x4 − 4x3

1. 
   finding intercepts, critical points, intervals of increase and decrease, points of inflection, intervals of concavity, local maximum or minimum points. If someone could give me just an example I could manage to follow that and figure them out please.

2. Homework Equations
?

3.
I attempted finding some of these analysis by putting these formulas in a graph but that's as far as I got.

 

[QuantumQuest]

Hello, I need to do a complete analysis using calculus and algebraic methods for these problems

1. ƒ(x) = −x2 + 4x − 3
2. g(x) = x3 − 6x2 + 9x
3. h(x) = x4 − 6x2
4. p(x) = x4 − 4x3

1. 
   finding intercepts, critical points, intervals of increase and decrease, points of inflection, intervals of concavity, local maximum or minimum points. If someone could give me just an example I could manage to follow that and figure them out please.

1. 
   I attempted finding some of these analysis by putting these formulas in a graph but that's as far as I got.

In order to understand what you have to do and why is it so, you have to study the respective theory of what you ask. There's no point to do things mechanically. So, what have you learned about all these you ask?

 

[PatrickrG]

In order to understand what you have to do and why is it so, you have to study the respective theory of what you ask. There's no point to do things mechanically. So, what have you learned about all these you ask?

in 1a. where f(x) is -x^2 + 4x -3 I plugged these into a graph and visually I see that x intercepts is x 0,-3 y is 1.5,0, no critical points, intervals of increase when x>0, decrease when x<0, no point of inflection/concavity or max/min. That is as far as I got

 

[QuantumQuest]

Why are you trying to figure out everything from the graph? This is a great way to verify things but not to calculate.

 

[PatrickrG]

Why are you trying to figure out everything from the graph? This is a great way to verify things but not to calculate.

I don't know how to use the calculus and algebraic methods unfortunately...

 

[QuantumQuest]

I don't know how to use the calculus and algebraic methods unfortunately...

OK, but is it required by your school to know them or you're self learning? I'm asking just to get some hint on how I can help in the best possible way.

 

[PatrickrG]

It is an online class It is required for that specific assignment, they want me to know different ways to solve things. Its hard for me because I learn better in a classroom setting where I can see examples and here it explained rather than online...But technically I am self learning because I have to wait quite sometime for an email back from the teacher for help so I'm stuck.

 

[QuantumQuest]

What is your background? Do you know how to solve a quadratic equation or higher degree equations? Do you know about derivatives? These are required in order to do a full study of a function.

 

[PatrickrG]

Been quite a few years since I completed high school and I'm taking this class because I need it so I can register for university. I don't know If I could solve a quadratic equation but have recently been learning about derivatives.

 

[QuantumQuest]

Been quite a few years since I completed high school and I'm taking this class because I need it so I can register for university. I don't know If I could solve a quadratic equation but have recently been learning about derivatives.

In this case you need to brush up by reading some introductory calculus text. Also some introductory linear algebra is needed. There is plenty of books for these. I recommend Stewart "Calculus" https://www.amazon.com/dp/0538497815/?tag=pfamazon01-20 and Gilbert Strang's "Introduction to Linear Algebra" https://www.amazon.com/dp/0980232716/?tag=pfamazon01-20. Khan Academy has also good introductory videos for these.

If you don't study and familiarize yourself with the required concepts, you won't build for yourself the required framework in order to go further.
It is no use to provide you with solutions, if you can't grasp what is going on. For instance, for the first function you ask, you must solve the quadratic equation ##-x^2 + 4x - 3 = 0## in order to find the x intercepts. You must take the discriminant ##\Delta = b^2 - 4ac## where a,b,c are the coefficients of the quadratic equation. It's sign shows if the quadratic has real solutions (##\Delta \geqslant 0##) or not (##\Delta < 0##). The roots are given by ##x = \frac{-b\pm\sqrt{\Delta}}{2a}##. These are the x - intercepts. Then you must take the first derivative of the function in order to find monotonicity (sign of ##f^\prime##) and the second derivative for points of inflection (##f^{\prime\prime} = 0##) etc. For the others you must make some factoring, take derivatives as before etc. So, as you can see, you must study in order to be able to do an analysis of a function.

 

[Mark44]

finding intercepts, critical points, intervals of increase and decrease, points of inflection, intervals of concavity, local maximum or minimum points. If someone could give me just an example I could manage to follow that and figure them out please.

It is an online class It is required for that specific assignment, they want me to know different ways to solve things. Its hard for me because I learn better in a classroom setting where I can see examples and here it explained rather than online...But technically I am self learning because I have to wait quite sometime for an email back from the teacher for help so I'm stuck.

Based on the portion of the problem statement that I quoted above, the expectation is that you are familiar with derivatives of functions, and can find them and use them to locate critical points, inflection points, and the other things asked for in this problem. If you are unfamiliar with these terms, you are probably in the wrong class.