# Is it possible to solve "minimum/maximum value of a function" problems using algebra?

• Jun 1st 2008, 08:09 PM
Chizum
Is it possible to solve "minimum/maximum value of a function" problems using algebra?
First off, let me say I know these problems can be solved with derivitives in calculus, but that's not what I'm after here.

For example:
What is the minimum value of the function f(x) = x^2 - 1?

I know the answer is is -1, but would constructing tables such as the one below be able to reliably bring me to the right answer for problems like this one? If yes, please elaborate the method you'd use when making these tables. I found this on the internet, but can't remember how this works from my Algebra 2 class which was years ago.
• Jun 1st 2008, 10:00 PM
mr fantastic
Quote:

Originally Posted by Chizum
First off, let me say I know these problems can be solved with derivitives in calculus, but that's not what I'm after here.

For example:
What is the minimum value of the function f(x) = x^2 - 1?

I know the answer is is -1, but would constructing tables such as the one below be able to reliably bring me to the right answer for problems like this one? If yes, please elaborate the method you'd use when making these tables. I found this on the internet, but can't remember how this works from my Algebra 2 class which was years ago.

The coordinates of the turning point of quadratic functin can be found by completing the square: y = a(x - h)^2 + k has a turning point at (h, k).

For your problem, h = 0 and k = -1 .......
• Jun 1st 2008, 11:43 PM
CaptainBlack
Quote:

Originally Posted by Chizum
First off, let me say I know these problems can be solved with derivitives in calculus, but that's not what I'm after here.

For example:
What is the minimum value of the function f(x) = x^2 - 1?

I know the answer is is -1, but would constructing tables such as the one below be able to reliably bring me to the right answer for problems like this one? If yes, please elaborate the method you'd use when making these tables. I found this on the internet, but can't remember how this works from my Algebra 2 class which was years ago.

This is an amplification of Mr Fantastics reply:

We wish to find the mininmum of a quadtratic objective function:

$\displaystyle f(x)=x^2+ax+b$

(we can work with monic quadratics as the objective:

$\displaystyle f_1(x)=kx^2+lx+m$

has its minima/maxima at the same point as:

$\displaystyle f_2(x)=x^2+\frac{l}{k}+\frac{m}{k}$

)

If we complete the square we have:

$\displaystyle f(x)=x^2+ax+b=(x+a/2)^2+b-\frac{a^2}{4}$

As $\displaystyle (x+a/2)^2 \ge 0$ this obviously has a minimum when $\displaystyle (x+a/2)^2=0$ which is when $\displaystyle x=-a/2.$

RonL
• Jun 2nd 2008, 01:05 PM
Mathstud28
Quote:

Originally Posted by Chizum
First off, let me say I know these problems can be solved with derivitives in calculus, but that's not what I'm after here.

For example:
What is the minimum value of the function f(x) = x^2 - 1?

I know the answer is is -1, but would constructing tables such as the one below be able to reliably bring me to the right answer for problems like this one? If yes, please elaborate the method you'd use when making these tables. I found this on the internet, but can't remember how this works from my Algebra 2 class which was years ago.

Now, I am not speaking in absolutes for there is an infinite amount of math I am unaware of. But to my knowledge the easiest way to generally find a max or min would be to apply Rolle's theorem well

Rolle's Theroem
$\displaystyle \text{Let f(x) be continous on [a,b] and differentiable}$$\displaystyle \text{ on (a,b), then if f(a)=f(b),}$$\displaystyle \text{ then there exists at least one point c on (a,b) such that f'(c)=0}$

Now you can adapt this to say has a relative max or min at c.

Then observing slopes on yoru interval (a,b) you can get a good estimate of your max or min,

This is just an approximatino of course, but it might be helpful if you are in a sticky situation