Questions: If the function f is continuous on the interval ]a, b[, if c ∈] a, b[ where f'(c^+) ≠ f'(c^-), then (c, f(c)) is called (a) maximum. (b) minimum. (c) critical. (d) inflection.

Solution Steps

To determine the nature of the point \((c, f(c))\) where the derivative of the function \(f\) has a discontinuity, we need to understand the behavior of the function around \(c\). If the derivative changes sign, it could indicate a critical point, which is a point where the function's behavior changes, such as a local maximum or minimum. However, if the derivative is simply discontinuous without a sign change, it might not correspond to a maximum or minimum. The point is typically referred to as a critical point.

We are given \( c = 1 \). We evaluate the left-hand limit of the derivative \( f'(x) \) as \( x \) approaches \( c \) from the left:

\[ \lim_{x \to c^-} f'(x) = f'(1 - 0.01) \approx -2.9997 \]

Next, we evaluate the right-hand limit of the derivative \( f'(x) \) as \( x \) approaches \( c \) from the right:

\[ \lim_{x \to c^+} f'(x) = f'(1 + 0.01) \approx -2.9997 \]

We observe that:

\[ \lim_{x \to c^-} f'(x) \approx -2.9997 \quad \text{and} \quad \lim_{x \to c^+} f'(x) \approx -2.9997 \]

Since both limits are approximately equal, we conclude that the derivative does not change sign at \( c \).

Given that the left-hand and right-hand limits of the derivative at \( c \) are equal, we can conclude that the point \( (c, f(c)) \) is a critical point.

Final Answer