(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that for all y(x)=ax^2+bx+c where a is a constant !=0 and x is a real number that [tex]\frac{y'(x2)^2- y'(x1)^2}{(x2-x1)}[/tex] = 2y''(x)

2. Relevant equations

I don't know what to put here in mathematics but here...

y(x)=ax^2+bx+c

y'(x)=2ax+b

y''(x)=2a

3. The attempt at a solution

I made a lot of attempts and tried many thing but here is the one that looks like it works.

y''(x)=dy'(x)/dx

y''(x) * dy/dx = y'(x)dy'/dx

now here I don't know if its ok to take the dx es of the fraction

y''(x) dy= y'(x)dy'

integral from x1 to x2 (y''(x) dy)= integral from x1 to x2 (y'(x)dy')

and since y''(x)=2a so its a constant

y''(x)* (x2-x1)= 1/2 * (y'(x2)^2-y'(x1)^2)

2y''(x)= (y'(x2)^2-y'(x1)^2)/ (x2-x1)

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# Homework Help: Proving an equation related to order of derivatives

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