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For any quadratic function f(x), the mean of the derivative of any two points is equal to the derivative of mean of those two points.

Let f(x) be a real valued quadratic function defined as:-

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

Then, f'(x)= 2ax+b

Let's consider a interval [i , j] that is defined under the domain of the function

Thus,

f'(i)=2ai+b . And

f'(j)=2aj+b

Then,

(f'(i)+f'(j))/2 = a(i+j)+b -(1)

Now, let x=(i+j)/2

f'(x)=f'((i+j)/2)=2a((i+j)/2)+b

= a(i+j)+b - (2)

From (1) & (2) we get

(f'(i)+f'(j) )/2 = f'((i+j)/2)