verify that the function satisfies the hypothesis of the mean value theorem on the given interval. then find all numbers c that satisfy the conclusion of the mean value theorem.
f(x) = x/(x+2) , [1,4]
f(b)-f(a)/b-a = f'(c) MVT
to find f(b) and f(a), just plug endpoints into original function
f(b) = f(4) = (2/3)
f(a) = f(1) = (1/3)
(2/3)-(1/3)
----------- = f'(c)
4 - 1
(1/9) = f'(c)
next, find derivative of f(x)
f'(c) = f'(x)
product rule
(1/9) = (x)(x+2)^-1
(1/9) = (x+2)^-1 - x(x+2)^-2
(1/9) = (1/x+2) - (x/(x+2))^2
(1/9) = (1/x+2) * (1 - (x/(x+2))
(1/x+2) = (1/9) mult. ea s. by 9
(9/x+2) = 1
9 = x + 2
7 = x
I'm sure you can solve the other x
To verify if a function satisfies the hypothesis of the Mean Value Theorem (MVT), we need to check two conditions:
1. Continuous on the given interval.
2. Differentiable on the open interval.
Let's go through each condition for the function f(x) = x/(x+2) on the interval [1, 4]:
1. Continuity: We need to check if the function is continuous on the closed interval [1, 4]. Since f(x) is a rational function, it is continuous everywhere except at values that make the denominator zero. In this case, the denominator (x+2) is non-zero for all x in the interval [1, 4]. Hence, the function is continuous on [1, 4].
2. Differentiability: We need to check if the function is differentiable on the open interval (1, 4). For f(x) to be differentiable, its derivative must exist on the open interval (1, 4). Let's find the derivative of f(x) and check if it exists on the open interval:
f(x) = x/(x+2)
To find the derivative, we can use the quotient rule:
f'(x) = (1*(x+2) - x*(1))/(x+2)^2
= 2/(x+2)^2
The derivative f'(x) = 2/(x+2)^2 exists for all x ≠ -2. Since -2 is not in the open interval (1, 4), the derivative exists on the open interval (1, 4), and the function f(x) is differentiable.
Therefore, the function f(x) = x/(x+2) satisfies the hypothesis of the Mean Value Theorem on the interval [1, 4].
Now let's find all the numbers c that satisfy the conclusion of the Mean Value Theorem. According to the Mean Value Theorem, there exists at least one number c such that:
f'(c) = (f(4) - f(1))/(4 - 1)
We can substitute the values of f(x) into the equation:
2/(c+2)^2 = (4/(4+2) - 1/(1+2))/3
Simplifying this equation, we have:
2/(c+2)^2 = (4/6 - 1/3)/3
= (2/3 - 1/3)/3
= 1/9
To solve for c, we can multiply both sides by (c+2)^2:
2 = 1/9 * (c+2)^2
Multiplying both sides by 9:
18 = (c+2)^2
Taking the square root of both sides:
±√18 = c + 2
Simplifying:
c + 2 = ±√18
Subtracting 2 from both sides:
c = -2 ±√18
Therefore, the conclusion of the Mean Value Theorem states that there exist two numbers c within the interval (1, 4) such that c = -2 + √18 and c = -2 - √18.