Final Answer:
By applying the mean value theorem to the function f(x)=x¹/5, There is no derivative at some point (option b).
Step-by-step explanation:
The mean value theorem states that if a function f(x) is continuous on a 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 instantaneous rate of change (derivative) equals the average rate of change. For the function f(x) = x^(1/5), it is continuous and differentiable on its domain. However, at x = 0, the derivative does not exist. Therefore, the mean value theorem cannot be applied at this point, and option b) "There is no derivative at some point" is correct (option b).
In the context of the given function, f(x) = x^(1/5), the derivative f'(x) can be calculated using the power rule. The derivative is f'(x) = (1/5)x^(-4/5). However, at x = 0, the derivative is undefined due to the presence of x^(-4/5). This lack of differentiability at x = 0 is a critical point where the mean value theorem cannot be applied, affirming option b) as the correct deduction.
Understanding the mean value theorem is essential for analyzing the behavior of functions and identifying points where certain conditions may not hold. In this case, recognizing the limitation of differentiability at x = 0 provides valuable insights into the characteristics of the given function.