Question

suppose f is known to be differentiable and contains a local extremum at c, then which of the following is true?
f(c) does not exist
f(c) < 0
f(c) = 0

Explanation:

Step1: Recall Fermat's Theorem

Fermat's Theorem states that if a function \( f \) is differentiable at a point \( c \) where it has a local extremum (local maximum or local minimum), then the derivative of \( f \) at \( c \), \( f'(c) \), must be equal to 0.

Step2: Analyze the given conditions

The problem states that \( f \) is differentiable at \( c \) and has a local extremum at \( c \). From Fermat's Theorem, we know that for a differentiable function at a local extremum point, the derivative at that point is zero.

• The first option says \( f'(c) \) does not exist, but the function is given to be differentiable at \( c \), so the derivative must exist. Thus, this option is false.
• The second option says \( f'(c) < 0 \), but at a local extremum of a differentiable function, the derivative is zero (not negative), so this option is false.
• The third option says \( f'(c) = 0 \), which is consistent with Fermat's Theorem for a differentiable function at a local extremum.

Answer:

\( f'(c) = 0 \) (the third option)