Left hand derivative and right hand derivative of a function f(x) at a point x=a are defined as:

• f'(a−) = lim h→0+ (f(a) − f(a−h)) / h
• f'(a+) = lim h→0+ (f(a+h) − f(a)) / h

Let f be a twice differentiable function. We also know that the derivative of an even function is an odd function and the derivative of an odd function is an even function.

The statement lim h→0 (f(−x) − f(−x−h)) / h = lim h→0 (f(x) − f(x−h)) / h implies that for all x ∈ R:

• A. f is odd
• B. f is even
• C. f is neither even nor odd
• D. Nothing can be concluded

To analyze the given statement, we need to understand the implications of the limits provided. The left-hand derivative at a point involves evaluating the function as we approach from the left, while the right-hand derivative involves approaching from the right. The equality of the two limits suggests a specific symmetry in the function.

Understanding Even and Odd Functions

An even function satisfies the condition f(-x) = f(x) for all x, while an odd function satisfies f(-x) = -f(x). The derivatives of these functions also follow specific patterns: the derivative of an even function is odd, and the derivative of an odd function is even.

Analyzing the Given Limit

The statement provided can be rewritten as:

• lim h→0 (f(−x) − f(−x−h)) / h = lim h→0 (f(x) − f(x−h)) / h

This implies that the behavior of the function f at -x mirrors its behavior at x. This symmetry indicates that f is likely an even function.

Conclusion on Function Type

Since the limit equality suggests that f(-x) behaves like f(x), we can conclude that:

• B. f is even

Thus, the correct answer is that f is an even function.