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The Mean Value Theorem is a fundamental concept in calculus and provides valuable information about a function's behavior between two points. It states that if a function \( g \) is continuous on a closed interval \([u, v]\) and differentiable on the open interval \((u, v)\), then there exists some point \( c \) in \((u, v)\) such that:\[g'(c) = \frac{g(v) - g(u)}{v - u}.\]This theorem ensures that there is at least one point where the instantaneous rate of change of the function (its derivative) is equal to the average rate of change over the interval.
The key is that the function must be both continuous and differentiable, making the theorem a powerful tool for understanding how functions behave.In the context of proving nondifferentiability, we use the Mean Value Theorem to investigate the derivative \( f' \). If we assume \( f \) is differentiable at a certain point \( a \), its derivative \( f' \) should be continuous.
Applying the Mean Value Theorem to \( f' \), we demonstrate how changes in the derivative’s direction lead to contradictions, proving that \( f \) cannot be differentiable at that point.

The First Derivative Test is a simple yet effective method to analyze extrema and intervals of increase or decrease for continuous functions. By examining the sign changes of the first derivative \( f'(x) \), we can deduce crucial information about the original function \( f(x) \).
Here’s how it works:

• If \( f'(x) \) changes from positive to negative at a point, this indicates a local maximum at that point.
• If \( f'(x) \) changes from negative to positive, there is a local minimum.
• If \( f'(x) \) does not change sign, \( f(x) \) has no extrema at that point.

Now, in the context of nondifferentiability, the First Derivative Test helps us observe the behavior of \( f' \) around the point \( a \). If \( f'(x) \) changes signs at \( a \), it suggests a sharp turn, indicating a potential point of nondifferentiability.
The combination of this test and subsequent analysis using the Mean Value Theorem solidifies the conclusion that \( f \) is not smooth or differentiable when both the first derivative and its change are scrutinized.

The Second Derivative Test takes the analysis one step further by utilizing the second derivative \( f''(x) \) to determine the nature of extrema more precisely. When \( f'(x) \) is known to be zero (a critical point).Here is how we employ the Second Derivative Test:

• If \( f''(x) > 0 \) at a critical point where \( f'(x) = 0 \), the function has a local minimum there.
• If \( f''(x) < 0 \), there is a local maximum.
• If \( f''(x) = 0 \), the test is inconclusive, and further investigation is required.

In tackling nondifferentiability, this test assists by further analyzing the concavity of \( f \) at point \( a \). The exercise challenges us to prove that \( f \) is nondifferentiable when \( f'' \) changes signs. This sign change indicates a reversal in concavity, suggesting a cusp or corner,
supporting the conclusion that differentiability is lost at \( a \). Thus, both the First and Second Derivative Tests, combined with the Mean Value Theorem, provide a robust method in assessing differentiability and understanding function behavior.