Proving the Non-one-to-one Property of f(x)=x^3-3x^2+2x on (-k,k)

[farmd684]

Homework Statement

show f(x)=x^3-3x^2+2x is not one to one on (-infinity,+infinity)

Homework Equations

finding the largest value of k such as f is one to one on interval (-k,k)

The Attempt at a Solution

i can get f`(x)=3x^2-6x+2 but it is positive so f(x) should be one-to-one
how to prove it

 

[Dick]

f(x)=3x^2-6x+2 isn't positive. f(1)=-1.

 

[HallsofIvy]

I just automatically looked at the discrimant for 3x²- 6x+ 2:
$$\sqrt{6^2- 4(3)(2)}= \sqrt{36- 24}= \sqrt{12}$$
Since that is a real number, the 3x²- 6x+2 changes sign, and the original function changes "direction", at two places.

 

[farmd684]

I just automatically looked at the discrimant for 3x²- 6x+ 2:
$$\sqrt{6^2- 4(3)(2)}= \sqrt{36- 24}= \sqrt{12}$$
Since that is a real number, the 3x²- 6x+2 changes sign, and the original function changes "direction", at two places.

thanks but how can i do this
largest value of k such as f is one to one on interval (-k,k)

 

[HallsofIvy]

A function f(x) is one-to-one as long as its derivative does not change sign- and a continuous derivative, such as the derivative of any polynomial, can change sign only where the derivative is 0.

Solve 3x²- 6x+ 2= 0, say by using the quadratic formula. Those 2 values will give 3 intervals on which the function is one to one. One of them contains the an interval of the form (-k, k).