# Finding Maxima/Minima of Polynomials without calculus?

• B
• PhotonSSBM
In summary, the conversation discusses the difficulty of finding the maxima and minima of polynomials without calculus or a calculator. The speaker suggests converting the polynomial into a form similar to that of a parabola to make it easier to find the maximum or minimum. They also mention that repeated roots in a polynomial can help determine the location of maxima and minima. There is also a discussion about how the students may be expected to reason about the signs of the factors in order to find the maxima and minima.
PhotonSSBM
I'm tutoring a student who is in a typical precalculus/trig course where they're teaching her about graphing various arbitrary polynomials. Among the rules of multiplicity and intercepts they seem to be phrasing the questions such that they expect the students to also find the maxima and minima of the polynomial as well. How do they expect their students to do this without calculus or the aid of a calculator? I was embarrassed sitting there with her mother not sure if I should teach the girl how to differentiate or not to answer the problem. But their the problem was, explicitly asking for the maxima of the function and saying no calculators allowed.

I'm stumped, is there anyone who knows what they're looking for?

For parabolas, you can convert them to the form f(x)=a(x-c)2+b where it is easy to find the maximum/minimum.
Sometimes higher order polynomials have similar expressions that allow finding the maximum/minimum without a derivative.

mfb said:
For parabolas, you can convert them to the form f(x)=a(x-c)2+b where it is easy to find the maximum/minimum.
Sometimes higher order polynomials have similar expressions that allow finding the maximum/minimum without a derivative.
Ah, good. I gave her the quick one for parabolas. But what you just said is interesting.

The forms of the polynomials were this:

##(x+3)^3(x-2)^2(x+7)^2##

Would there be some way to find the maximum or minimum given this form already?

Yes, with the same idea as for the parabola. You only get local maxima and minima here and you'll miss half of them.

PhotonSSBM said:
The forms of the polynomials were this:

##(x+3)^3(x-2)^2(x+7)^2##

You can see from calculus that if a polynomial has a "repeated root" then that value is also a root of its derivative.

Perhaps the students are expected to reason about the signs of the factors. For example, the factor (x-2) changes signs "as x changes from less than 2 to greater than 2", but since that factor is squared, the polynomial doesn't change sign.

Stephen Tashi said:
but since that factor is squared, the polynomial doesn't change sign.
Which also means there is a maximum or minimum at that zero.

## 1. What is the process for finding the maximum or minimum value of a polynomial without using calculus?

The process for finding the maximum or minimum value of a polynomial without using calculus is to first determine the degree of the polynomial, which is the highest exponent of the variable. Then, use the leading coefficient and the degree of the polynomial to determine the end behavior of the graph. Finally, use the techniques of factoring or completing the square to find the x-coordinate of the vertex, which will give the maximum or minimum value.

## 2. Can a polynomial have more than one maximum or minimum value?

Yes, a polynomial can have more than one maximum or minimum value. This occurs when the graph of the polynomial has multiple peaks or valleys. The number of maximum or minimum values a polynomial can have is equal to its degree minus one.

## 3. How do you know if a polynomial has a maximum or minimum value?

A polynomial has a maximum or minimum value when its graph has a turning point, also known as a vertex. In other words, the slope of the graph changes from positive to negative (maximum) or negative to positive (minimum). This can also be determined by analyzing the end behavior of the graph.

## 4. Is there a shortcut or formula for finding the maximum or minimum value of a polynomial?

There is no specific formula for finding the maximum or minimum value of a polynomial without using calculus. However, there are techniques such as factoring or completing the square that can help make the process more efficient. It is also helpful to have a good understanding of the properties and behavior of polynomial graphs.

## 5. Can calculus be used to find the maximum or minimum value of a polynomial?

Yes, calculus can be used to find the maximum or minimum value of a polynomial. The derivative of a polynomial can be used to find the critical points, which are points where the slope of the graph is equal to 0. The x-coordinate of these points will give the maximum or minimum value of the polynomial. However, this method requires knowledge of calculus and may not be accessible to those without a background in the subject.

Replies
11
Views
2K
Replies
1
Views
2K
Replies
2
Views
3K
Replies
45
Views
5K
Replies
9
Views
6K
Replies
2
Views
2K
Replies
3
Views
2K
Replies
7
Views
2K
Replies
6
Views
5K
Replies
4
Views
4K