Finding Local Maxima and Minima Points for a Cubic Function

In summary, the local max and min points for g(x) can be found using Newton's method and the cubic formula.
  • #1
prophet05
12
0
[SOLVED] Local Max & Min problem

I'm having a real tough time finishing this problem. I have to find the local maxima and minima points for g(x).

g(x) = 1 + 4x - 10x^2 + x^4

(dy/dx) = 4 - 20x + 4x^3

I've been trying to factor it to get the max and min points, but I find it impossible to simplify. I've tried factoring and just can't seem to find a way. And I can't use quadratic formula since it's to the 3rd degree, right?
 
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  • #2
The first thing I thought of was Newton's method because I'm not great when it comes to solving the zeros for that with pure algebra.
 
  • #3
Well here's the cubic formula:

cubic.gif


For a cubic of the form ax^3+bx^2+cx+d

The roots are: 0.2016396757234, 2.12841906384458, -2.33005873956798

So Newton's method seems like that way to go.
 
  • #4
Yea, the cubic formula is out, I am not going to mesmerize that. I'll try and read up on Newton's method.
 
  • #5
Newton's method is a powerful tool for finding zeros. Take a guess, x_n, at what x-value the zero could be at:

Then [tex]x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}[/tex]

Then take the value of [tex]x_{n+1}[/tex] that you got and do the above calculation over and over again (around four or five times) until the number that you obtain settles down.

The only problem is that it doesn't tell you how many zeros the function has.
 
  • #6
"Mesmerize" it? Now why didn't I think of that! I'll just hynotize formulas into doing my bidding!
 
  • #7
HallsofIvy said:
"Mesmerize" it? Now why didn't I think of that! I'll just hynotize formulas into doing my bidding!

It's all part of the plan for world domination mwhahahaha!11!oneone!1!1
 
  • #8
Feldoh said:
It's all part of the plan for world domination mwhahahaha!11!oneone!1!1

I don't usually jump on the bandwagon of a string of jokes, but that 'mwahahaha!11!oneone... had me rofl-ing!
 
  • #9
Haha. Thanks for all the help. Turns out the professor wanted us to solve it with calculator. Mesmerize would have been a better way to go. =P

[tex]g(x) = 1 + 4x - 10x^2 + x^4[/tex]
[tex](dy/dx) = 4 - 20x + 4x^3[/tex]
 
Last edited:

What is a "Local Max & Min problem" in the context of mathematics?

A "Local Max & Min problem" is a type of mathematical problem that involves finding the highest and lowest points, or peaks and valleys, in a given function within a specific interval. These points are known as local maxima and minima, and they represent the highest and lowest values of a function within a specific range of inputs.

How is a "Local Max & Min problem" different from a "Global Max & Min problem"?

While a "Local Max & Min problem" focuses on finding the highest and lowest points within a specific interval, a "Global Max & Min problem" seeks to find the overall highest and lowest values of a function across its entire domain. This means that a global maximum or minimum may not necessarily be a local maximum or minimum, and vice versa.

What is the importance of solving "Local Max & Min problems"?

Solving "Local Max & Min problems" is important in mathematics because it helps us understand the behavior of a function within a specific interval. This information can be used to optimize the function for a specific purpose, such as finding the maximum profit for a business or the minimum cost for a production process.

What are some common methods for solving "Local Max & Min problems"?

One common method for solving "Local Max & Min problems" is the first derivative test, where the critical points of a function are found by setting the first derivative equal to zero and examining the sign of the second derivative. Another method is the second derivative test, which uses the second derivative to determine whether a critical point is a local maximum, minimum, or neither.

Can "Local Max & Min problems" have multiple solutions?

Yes, "Local Max & Min problems" can have multiple solutions, especially when the function is complex or has multiple local maxima and minima within the given interval. It is important to carefully examine the behavior of the function and consider all possible critical points to identify all potential solutions.

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