Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval?
f(x)= ln(x) , [1,6]
If it satisfies the hypotheses, find all numbers c that satisfy the conclusion of the Mean Value Theorem. (Enter your answers as a comma-separated list. If it does not satisfy the hypotheses, enter DNE).
try :
https://www.google.com/search?q=graph+ln%28x%29+from+1+to+6&ie=utf-8&oe=utf-8&aq=t&rls=org.mozilla:en-US:official&client=firefox-a&channel=sb
ln 6 = 1.79
ln 1 = 0
ln 6 - ln 1 = 1.79
1.79/5 = .36 approx
ln x = .36
e^ln x = ? = e^.36
so
x = 1.43 sure enough between 1 and 6 :)
To determine if the function satisfies the hypotheses of the Mean Value Theorem on the interval [1, 6], we need to check the following conditions:
1. Continuity: The function f(x) = ln(x) is continuous on the closed interval [1, 6]. This condition is satisfied because ln(x) is a continuous function for all x > 0.
2. Differentiability: The function f(x) = ln(x) is differentiable on the open interval (1, 6). To confirm this, we need to check if the derivative exists for all x in the interval (1, 6).
The derivative of ln(x) is given by d/dx ln(x) = 1/x. This derivative exists for all x > 0, so the function is differentiable on the interval (1, 6).
Now, since the function satisfies both conditions of the hypotheses, we can proceed to find the numbers c that satisfy the conclusion of the Mean Value Theorem.
According to the Mean Value Theorem, if a function f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one number c in (a, b) such that f'(c) = (f(b) - f(a))/(b - a).
In this case, a = 1, b = 6, and f(x) = ln(x). So, we need to find the value of c that satisfies f'(c) = (f(6) - f(1))/(6 - 1).
To do this, we first calculate f'(x) = 1/x. Now we can solve the equation 1/c = (ln(6) - ln(1))/(6 - 1) = ln(6)/5 to find the value of c.
To solve for c, we can cross-multiply and simplify:
1 = (ln(6)/5) * c
c = 5/ln(6)
Therefore, the number that satisfies the conclusion of the Mean Value Theorem for the given function on the interval [1, 6] is c = 5/ln(6).