Finding Functions from Derivatives


Suppose that f(−1) = 3 and that f'(x) = 0 for all x. Must f(x) = 3 for all x? Give reasons for your answer.

Verified step by step guidance

1. First, understand the meaning of the derivative f'(x) = 0 for all x. This implies that the function f(x) has a constant slope of 0, meaning it is a horizontal line.
2. Since the derivative represents the rate of change of the function, a derivative of 0 indicates that the function does not change as x changes. Therefore, f(x) must be a constant function.
3. Given that f(x) is a constant function, it must take the same value for all x.
4. We are provided with the information that f(−1) = 3. Since f(x) is constant, this value must be the same for all x.
5. Thus, we conclude that f(x) = 3 for all x, because a constant function with f(−1) = 3 implies that the constant value is 3 everywhere.

Key Concepts

Constant Function

A constant function is a function that always returns the same value, no matter the input. In mathematical terms, if f'(x) = 0 for all x, it implies that the function f(x) does not change as x changes, indicating that f(x) is a constant function.

Derivative

The derivative of a function at a point measures the rate at which the function's value changes as its input changes. If the derivative f'(x) is zero for all x, it means the function has no slope and is flat, suggesting that the function is constant across its domain.

Initial Condition

An initial condition provides specific information about a function at a particular point, which helps determine the constant of integration when solving differential equations. In this problem, f(−1) = 3 serves as an initial condition, confirming that the constant value of the function f(x) is 3.