Find a point c satisfying the conclusion of the Mean Value Theorem for the following function and interval.
f(x)=x^{- {6}}, \qquad[1, 2]
c =
To find a point c satisfying the conclusion of the Mean Value Theorem for the function f(x) = x^(-6) on the interval [1, 2], we need to check if the conditions of the theorem are met.
The Mean Value Theorem states that 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 point c in (a, b) where the slope of the tangent line at c is equal to the average rate of change of f(x) over the interval [a, b]. In equation form, it is expressed as:
f'(c) = (f(b) - f(a))/(b - a)
Let's first calculate the average rate of change of f(x) over the interval [1, 2]:
f(2) = 2^(-6) = 1/64
f(1) = 1^(-6) = 1
So, the average rate of change (f(b) - f(a))/(b - a) is:
(1/64 - 1)/(2 - 1) = (-63/64)/1 = -63/64
Next, we need to find the derivative of f(x):
f'(x) = -6x^(-7)
Now we can set up the equation f'(c) = -63/64 and try to solve for c:
-6c^(-7) = -63/64
To simplify the equation, we can multiply both sides by 64 to get rid of the fraction:
-384c^(-7) = -63
Now, divide both sides by -384:
c^(-7) = -63/-384
Simplifying the right side:
c^(-7) = 63/384
c^(-7) = 7/48
To find c, we need to take the seventh root of both sides:
c = (7/48)^(1/7)
Using a calculator, we can evaluate this expression to find the numerical value of c:
c ≈ 1.070
Therefore, a point c ≈ 1.070 satisfies the conclusion of the Mean Value Theorem for the function f(x) = x^(-6) on the interval [1, 2].
To find the point c that satisfies the conclusion of the Mean Value Theorem for the function f(x) = x^(-6) on the interval [1, 2], we need to verify the conditions of the theorem.
The Mean Value Theorem states that if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a point c in (a, b) where the derivative of the function is equal to the average rate of change of the function over that interval.
In this case, the function f(x) = x^(-6) is continuous on the closed interval [1, 2] and differentiable on the open interval (1, 2).
Now, let's find the derivative of f(x) using the power rule:
f'(x) = -6x^(-7)
Next, we need to find the average rate of change of f(x) over the interval [1, 2]. The average rate of change is given by the formula:
average rate of change = (f(2) - f(1))/(2 - 1)
Plugging in the values of f(2) and f(1):
average rate of change = (2^(-6) - 1^(-6))/(2 - 1)
= (1/64 - 1)/(1)
= (1 - 1)/(64)
= 0/64
= 0
Since the average rate of change is 0, we need to find a point c in the interval (1, 2) where the derivative f'(x) is also equal to 0.
Solving f'(c) = 0:
-6c^(-7) = 0
Dividing both sides by -6:
c^(-7) = 0
Taking the reciprocal of both sides:
(c^(-7))^(-1) = 0^(-1)
c^7 = undefined
Therefore, there is no point c in the interval (1, 2) where the derivative f'(x) is equal to the average rate of change of f(x) over the interval.
Hence, we cannot find a point c that satisfies the conclusion of the Mean Value Theorem for the function f(x) = x^(-6) on the interval [1, 2].