Devin's Reflection

Mean Value Theorem
If f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a number c in (a,b) such that

f'(c) = f(b) - f(a)/b-a

The "mean" in the Mean value Theorem refers to the mean rate of change of f in the interval [a,b]. In the equation y = f(1) - f(2)/1-2(x-2) - f(2)
Let g(x) = f(x) - y
So it would be f(x) - the equation
By evaluating g at a and b, you can see that g(a) = 0 = g(b). Because f is continuous on [a,b], it follows that g is also continuous on [a,b]. Furthermore, because f is differentiable, g is also differentiable, and you can apply Rolle's Theorem to the function g. So, there exists a number c in (a,b) such that g'(c) = 0, which implies that function g. So, there exists a number c (a,b) such that g'(c) = 0, which implies that 0 = g'(c)
= f'(c) - f(1) - f(2)/1-2
So, there exists a number c in (a,b) such that
f'(c) = f(1) - f(2)/1-2