Determining increase/decrease intervals for ax^2+bx+c

[peripatein]

Hello,

Homework Statement

I am trying to determine on what intervals the parabola y=ax^2+bx+c increases/decreases, without resorting to differentiation.

Homework Equations

The Attempt at a Solution

For the function to be increasing on a certain interval f(x1)>f(x2) for any x1 and x2 on that interval such that x1>x2. Hence, ax1^2+bx1+c>ax2^2+bx2+c. That yields, (x1+x2)>-b/a. How do I derive the expected x>-b/2a from that? That result could be obtained for x1=x2, hence Δ=0, but why is that and why ought it to be used in order to obtain the correct answer?

 

[mfb]

Let x2=x1+Δ and then consider the limes of Δ->0?

 

[Dick]

Hello,

Homework Statement

I am trying to determine on what intervals the parabola y=ax^2+bx+c increases/decreases, without resorting to differentiation.

Homework Equations

The Attempt at a Solution

For the function to be increasing on a certain interval f(x1)>f(x2) for any x1 and x2 on that interval such that x1>x2. Hence, ax1^2+bx1+c>ax2^2+bx2+c. That yields, (x1+x2)>-b/a. How do I derive the expected x>-b/2a from that? That result could be obtained for x1=x2, hence Δ=0, but why is that and why ought it to be used in order to obtain the correct answer?

How about completing the square?