Finding Maxima/Minima of Polynomials without calculus?

[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?

 

[mfb]

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.

 

[PhotonSSBM]

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?

 

[mfb]

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

 

[Stephen Tashi]

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.

 

[mfb]

but since that factor is squared, the polynomial doesn't change sign.

Which also means there is a maximum or minimum at that zero.