Final answer:
To prove that the function does not have a local maximum nor a local minimum, we need to analyze the behavior of its derivative. By finding the derivative of the function and identifying its critical points, it can be shown that the function does not have any local extrema.
Step-by-step explanation:
In order to determine if the function f(x) = x101 + x51 + x + 1 has neither a local maximum nor a local minimum, we need to analyze the behavior of the function's derivative. The derivative of f(x) can be found by taking the derivative of each term separately, which gives us f'(x) = 101x100 + 51x50 + 1.
To find the critical points, we need to solve the equation f'(x) = 0. However, since the derivative is always positive or always negative (for any x value), there are no critical points. This means that the function does not have any local extrema, and therefore, neither a local maximum nor a local minimum.