Given f(x)=4+3/x find all values of c in the interval (1,3) that satisfy the mean value theorem.
A. 2
B. Sqrt(2)
C. Sqrt(3)
D. +or- sqrt(3)
E. MVT doesn't apply
I got C
Hmm...
Mean value says that
(f(x2)-f(x1))/(x2-x1) = f(x0)
where x0∈[x1,x2]
(f(x2)-f(x1))/(x2-x1) = (5-7)/2 = -1
f'(x)=-3/x^2
f'(sqrt(3))=-3/3=-1,
Yes, C is correct.
To check which values of c satisfy the mean value theorem for the function f(x)=4+3/x in the interval (1,3), we need to verify if the average rate of change of the function on that interval is equal to the instantaneous rate of change at some point within it.
The mean value theorem states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one value c in the interval (a, b) such that the instantaneous rate of change at c is equal to the average rate of change over the interval [a, b].
To begin, let's find the average rate of change over the interval (1,3):
Average rate of change = (f(3) - f(1))/(3 - 1)
Substituting the function f(x)=4+3/x:
Average rate of change = (4+3/3 - (4+3/1))/(3 - 1)
= (4+1 - 4-3)/(3 - 1)
= (5 - 7)/(2)
= -2/2
= -1
Next, let's find the derivative of f(x) with respect to x:
f'(x) = d/dx (4+3/x)
= 0 - 3/x^2
= -3/x^2
Now, we need to find the instantaneous rate of change at some point c within the interval (1,3). We can set the derivative equal to the average rate of change we found earlier:
-3/c^2 = -1
Simplifying the equation:
-3 = -c^2
Rearranging:
c^2 = 3
Taking the square root of both sides:
c = ±√3
Therefore, the values of c that satisfy the mean value theorem in the interval (1,3) are ±√3.
Hence, the correct answer is D. ±√3.
To find the values of c that satisfy the mean value theorem for the function f(x)=4+3/x in the interval (1,3), we need to follow these steps:
1. Confirm that the function f(x) is continuous on the closed interval [1, 3]. In this case, f(x) is defined for all values of x except x = 0. Since 0 is not in the interval (1, 3), we can conclude that f(x) is continuous on [1, 3].
2. Calculate the average rate of change of f(x) over the interval [1, 3]. The average rate of change, denoted as f'(c), can be calculated using the formula:
f'(c) = (f(3) - f(1))/(3 - 1)
Substituting the function f(x) into the formula, we have:
f'(c) = (4+3/3 - (4+3/1))/(3 - 1)
Simplifying, we get:
f'(c) = (4+1 - 7)/(2)
= (5 - 7)/2
= -2/2
= -1
3. We need to solve for c in the equation f'(c) = -1. Substituting the function f(x), we have:
-1 = 3/c^2
Rearranging the equation, we get:
3 = -c^2
Simplifying further, we have:
c^2 = -3
Since the square of any real number is always positive or zero, we cannot find a real number c that satisfies the equation c^2 = -3. Therefore, the mean value theorem does not apply in this case.
The answer is E. MVT doesn't apply.