Suppose f(x)=αx^2+βx+γ
is a general quadratic polynomial with α≠0 and let [a,b] be any interval. According to the Mean Value Theorem there is at least one number c such that f(b)−f(a)=f′(c)(b−a).
In this particular case the number c is unique and it's independent of the coefficients of f.
c=____
Your answer will be in terms of a and b.
see the related questions below.
To find the value of c, let's start by finding f'(x), the derivative of f(x):
f(x) = αx^2 + βx + γ
To find f'(x), we apply the power rule for differentiation. Differentiating each term will give us:
f'(x) = 2αx + β
Now, according to the Mean Value Theorem, there exists a number c such that:
f(b) - f(a) = f'(c)(b - a)
Substituting the values into the equation, we have:
f(b) - f(a) = (2αc + β)(b - a)
To determine the value of c, we can rearrange the equation:
(2αc + β)(b - a) = f(b) - f(a)
Expanding the equation:
2αc(b - a) + β(b - a) = f(b) - f(a)
Simplifying the equation:
2αc(b - a) = f(b) - f(a) - β(b - a)
Now, we can isolate c by dividing both sides of the equation by 2α(b - a):
c = [f(b) - f(a) - β(b - a)] / (2α(b - a))
Therefore, c in terms of a and b is:
c = [f(b) - f(a) - β(b - a)] / (2α(b - a))
To find the value of c in terms of a and b, we need to apply the Mean Value Theorem to the given quadratic function f(x) = αx^2 + βx + γ on the interval [a, b].
First, let's find the derivative of the function f(x):
f'(x) = 2αx + β
According to the Mean Value Theorem, there exists at least one number c in the interval [a, b] such that f(b) - f(a) = f'(c)(b - a).
Now, let's substitute the values into this equation:
f(b) - f(a) = (2αc + β)(b - a)
We want to find the value of c, so isolating it in the equation:
f(b) - f(a) = (2αc + β)(b - a)
Divide both sides of the equation by (2α(b - a)):
(f(b) - f(a)) / (2α(b - a)) = c
Therefore, the number c in terms of a and b is:
c = (f(b) - f(a)) / (2α(b - a))
Note: The value of c based on the Mean Value Theorem for this particular case is unique and independent of the coefficients α, β, and γ of the quadratic function f(x).