FIRST: Highest & Lowest Root of f(x) = x^2 - 2x + 1

[tomcenjerrym]

FIRST
What is the HIGHEST root? If there is a highest root, are there available any LOWEST root? Say, what is the highest and lowest root of f(x) = x^2 - 2x + 1?

SECOND
What happen if I sum the EVEN and ODD function? I don’t think I am a good on geometry.

THIRD
What is the integral of \frac {1}{x^2} or \int\frac{1}{x^2}?

Is it \frac{x^{-2 + 1}}{-2+1} + C?

Can I make it in natural logarithm ln form?

 

[robert Ihnot]

Three you got right. As for One: (X-1)^2, what do you mean highest or lowest?

 

[tomcenjerrym]

Sorry for not being careful. Now I understand what is meant by HIGHEST and LOWEST root. Say, for the following equation:

f(x) = x^2 - x - 6

The highest root is x = 3 and lowest is x = -2.

In case of f(x) = x^2 - 2x + 1 or f(x) = (x - 1)^2 there is no highest and lowest root because the only available root is x = 1.

Correct me if I am wrong.

Thanks

 

[matt grime]

Largest and smallest, not highest and lowest.

 

[tomcenjerrym]

Why? Is it about interval?

 

[matt grime]

No, it just a matter of the meaning of the words in English. If you talk about the highest root no one will know for sure what you mean. The largest root is better.

 

[HallsofIvy]

FIRST
What is the HIGHEST root? If there is a highest root, are there available any LOWEST root? Say, what is the highest and lowest root of f(x) = x^2 - 2x + 1?

Of a specific equation? If by "highest" you mean largest and by "lowest" you mean smallest, and the equation has a finite number of roots, then yes, it must have a largest and a smallest root. It happens that the polynomial you give x²- 2x+1= (x-1)² has only a single root, x= 1, so that is both the largest and smallest root is 1.
On the other hand, the equation x²+ 1= 0 has only imaginary roots and so does not have a "largest" and "smallest" root- the field of complex numbers cannot be ordered.
Also, the equation sin(x)= 0 has an infinite number of roots, any multiple of \pi and so does not have either a "largest" or "smallest" root.

(By the way, strictly speaking, an equation has "roots". A function has "zeroes": the roots of the equation f(x)= 0. But people are seldom that strict!)

SECOND
What happen if I sum the EVEN and ODD function? I don’t think I am a good on geometry.

What do you mean by "the" even and "the" odd functions? If you mean add an arbitrary odd and an arbitrary even function, you get a function that is neither even nor odd. Functions that are neither even nor odd are far more common than even or odd functions.

Have you considered the possibility that a function may be BOTH even and odd? It is possible!

THIRD
What is the integral of \frac {1}{x^2} or \int\frac{1}{x^2}?

Is it \frac{x^{-2 + 1}}{-2+1} + C?

Can I make it in natural logarithm ln form?

You don't need natural logarithm: your first formula is correct. you have x^-2 so you use "int of xⁿ is \frac{x^{n+1}}{n+1} as long as n+1 is not 0". The only time you need natural logarithm is when n+1= 0- in other words, when n= -1.