Analysis: Proving a unique mean value property of quadratic functions a. Show  $ quadratic polynomial $f(x)=ax^2+bx+c $, point c in Mean Value is midpoint.
For a: I know Mean Value Theorem states that (conditions being satisfied f(x) is continuous on $[g,h]$, f(x) is differentiable on $(g,h)$.  $f'(c)=f(h)-f(g)/(h-g)$.  So I take derivative of our function and get $2ax+b$.  $f'(c)=2ac+b$.  Thus we have $2ac+b=f(h)-f(g)/(h-g)$.  Solving for c we get: $c=((f(h)-f(g)/(h-g)) -b)/2a$  Plugging in, we get: $((ah^2+bh+c)-(ag^2+bg+c)/(h-g)-b)/2a$.   And,I'm not getting c to be the midpoint, so I think I'm doing something completely wrong.  
 A: $$f'(c)=b+2ac$$
$$\frac{f(h)-f(g)}{h-g}=\frac{-a g^2+a h^2-b g+b h}{h-g}$$
$$b+2ac=\frac{-a g^2+a h^2-b g+b h}{h-g}$$
$$2ac=\frac{-a g^2+a h^2-b g+b h}{h-g}-b$$
$$2ac =\frac{-a g^2+a h^2-b g+b h-bh+bg}{h-g}$$
$$c=\frac{a(h+g)(h-g)}{2a(h-g)}$$
$$c=\frac{h+g}{2}$$
Hope this helps
A: a)
$f'(x) = 2ax  + b$
MVT: There exists $c \in (g,h)$ such that: $f'(c) = \frac {f(g) - f(h)}{g-h}$
$\frac {f(g) - f(h)}{g-h}$ simplifies to:
$\frac {ag^2 + bg + c - ah^2 - bh  - c}{g-h}\\
\frac {a(g^2 - h^2) + b(g - h)}{g-h}\\
\frac {a(g - h)(g+h) + b(g - h)}{g-h}\\
a(g+h)$ 
$f'(x) = 2ac + b = a(g+h) + b\\
c = \frac {g+h}{2}$
A: If $f(x)=ax^2+bx+c$ then $f'(x)=2ax+b$. It follows that
$$f(y)-f(x)=a(y^2-x^2)+b(y-x)=\left(2a{x+y\over2}+b\right)(y-x)=f'\left({x+y\over2}\right)(y-x)\ .$$
This proves that quadratic polynomials indeed have the alleged property. 
Now the converse: Assume that a function $f:\>{\mathbb R}\to{\mathbb R}$ satisfies
$$f(t+h)-f(t-h)=2h f'(t)\tag{1}$$
for all $t$ and all $h$. Since $f$ is differentiable it follows that $f'$ is differentiable as well, hence $f''(t)$ exists for all $t$. If we differentiate $(1)$ with respect to $h$ for fixed $t$ we obtain
$$f'(t+h)+f'(t-h)=2f'(t)\ ,$$
and differentiating agin with respect to $h$ gives
$$f''(t+h)-f''(t-h)=0\qquad \forall t, \  \forall h\ .$$
The last identity is saying that $f''$ is constant. It follows that any $f$ satisfying $(1)$ has to be a quadratic polynomial.
