Answer:
the value of c that satisfies the conclusion of the mean value theorem is c = 5
Explanation:
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 number c in the interval (a, b) such that:
f'(c) = (f(b) - f(a)) / (b - a)
In this case, we are given the function f(x) = 10x, and the interval [a, b] is [0, 2]. Plugging these values into the equation above, we get:
f'(c) = (f(2) - f(0)) / (2 - 0)
= (20 - 0) / 2
= 10 / 2
= 5
Therefore, the value of c that satisfies the conclusion of the mean value theorem is c = 5.
UPDATED: To find the value of c that satisfies the conclusion of the mean value theorem for the function f(x) = 10^x over the interval [0, 2], you can use the equation:
f'(c) = (f(b) - f(a)) / (b - a)
In this case, a = 0, b = 2, and f(x) = 10^x, so we can plug these values into the equation to find the value of c:
f'(c) = (f(2) - f(0)) / (2 - 0)
= (10^2 - 10^0) / 2
= 100 / 2
= 50
Therefore, the value of c that satisfies the conclusion of the mean value theorem is c = 50.
I hope this helps clarify things. If you have any further questions, please don't hesitate to ask.