18) Find all numbers c that satisfy the conclusion of the Mean Value Theorem for the following function and interval. Enter the values in increasing order and enter N in any blanks you don't need to use.
f(x)=2x/6x+12,[1,4]
f(x) = 2x/(6x+12)
f(1) = 1/9
f(4) = 2/9
(f(4)-f(1))/(4-1) = 1/27
The MVT says that there is c in [1,4] such that f'(c) = 1/135
f'(x) = 2/3 (x+2)^-2
So,
2/3 (c+2)^-2 = 1/27
c = 2.24
which is inside [1,4]
To find all numbers c that satisfy the conclusion of the Mean Value Theorem for the given function f(x) = 2x/(6x + 12) on the interval [1, 4], we need to check if the function meets the criteria of the 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), there exists at least one number c in the open interval (a, b) such that the derivative of f(c) is equal to the average rate of change of f(x) over the interval [a, b].
Step 1: Check the continuity of f(x) on the interval [1, 4].
The function f(x) = 2x/(6x + 12) is a rational function, and it is continuous for all values of x except where the denominator is equal to 0. In this case, the denominator 6x + 12 = 0 when x = -2. Therefore, f(x) is continuous on the interval [1, 4].
Step 2: Check the differentiability of f(x) on the interval (1, 4).
To check the differentiability, we need to find the derivative of f(x) and ensure that it is defined and continuous on the open interval (1, 4).
Let's find the derivative of f(x):
f'(x) = (2*(6x + 12) - 2x*6)/(6x + 12)^2
= (12x + 24 - 12x)/(6x + 12)^2
= 24/(6x + 12)^2
The derivative f'(x) is defined and continuous for all values of x except where the denominator (6x + 12)^2 is equal to 0. The denominator is never equal to 0, so f'(x) is defined and continuous on the interval (1, 4).
Since f(x) is continuous on [1, 4] and differentiable on (1, 4), we can apply the Mean Value Theorem to find the value of c.
Step 3: Calculate the average rate of change of f(x) over the interval [1, 4].
The average rate of change of f(x) over [1, 4] can be found by evaluating (f(4) - f(1))/(4 - 1).
f(4) = 2*4/(6*4 + 12) = 8/36 = 2/9
f(1) = 2*1/(6*1 + 12) = 2/18 = 1/9
The average rate of change = (2/9 - 1/9)/(4 - 1) = 1/9
Step 4: Solve for c.
We need to solve for c using the equation f'(c) = average rate of change obtained in step 3.
f'(c) = 24/(6c + 12)^2 = 1/9
Now, we can solve this equation to find the values of c.
24/(6c + 12)^2 = 1/9
Multiply both sides by 9:
216 = (6c + 12)^2
Take the square root of both sides:
±√216 = 6c + 12
We can simplify √216 to ±6√6.
±6√6 = 6c + 12
Subtract 12 from both sides:
±6√6 - 12 = 6c
Divide both sides by 6:
±√6 - 2 = c
So, the values of c that satisfy the conclusion of the Mean Value Theorem for the function f(x) = 2x/(6x + 12) on the interval [1, 4] are ±√6 - 2.
To find the numbers 'c' that satisfy the conclusion of the Mean Value Theorem for the function f(x) = (2x)/(6x + 12) on the interval [1, 4], we need to follow these steps:
1. Calculate the average rate of change of the function over the interval [1, 4].
To find the average rate of change, we need to evaluate f(b) - f(a) / (b - a), where 'a' and 'b' are the endpoints of the interval.
So, we have:
f(4) = (2 * 4) / (6 * 4 + 12) = 8 / 36 = 2 / 9
f(1) = (2 * 1) / (6 * 1 + 12) = 2 / 18 = 1 / 9
The average rate of change is:
(2 / 9) - (1 / 9) / (4 - 1) = 1 / 9 / 3 = 1 / 27
2. Find the derivative of the function.
To apply the Mean Value Theorem, we need to find the derivative of f(x).
To calculate the derivative, we use the quotient rule:
f'(x) = (2 * (6x + 12) - 2x * 6) / (6x + 12)^2
= (12x + 24 - 12x) / (6x + 12)^2
= 24 / (6x + 12)^2
= 4 / (x + 2)^2
3. Find the values of 'c' where f'(c) equals the average rate of change.
Since f'(c) must be equal to 1/27, we set 4 / (c + 2)^2 = 1/27 and solve for 'c'.
4 / (c + 2)^2 = 1/27
Multiply both sides by (c + 2)^2 to eliminate the fraction:
4 = (c + 2)^2 / 27
Multiply both sides by 27 to get rid of the fraction:
108 = (c + 2)^2
Take the square root of both sides:
±√108 = ±(c + 2)
±3√3 = ±(c + 2)
If we solve the equation c + 2 = 3√3, we get c = 3√3 - 2.
If we solve the equation c + 2 = -3√3, we get c = -3√3 - 2.
So, the values of 'c' that satisfy the conclusion of the Mean Value Theorem are:
c = 3√3 - 2 and c = -3√3 - 2.