Question

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f^{\prime}(x)=0\) for all \(x\) in the domain of \(f,\) then \(f\) is a constant function.

Short answer

The statement is false. Even though a function like \(|x|\) has its derivative equal to zero at some points, it is not a constant function. The assertion would be true if the interval was specific and not the entire domain of the function. Hence, the derivative of a function being zero for all points in its domain does not necessarily imply that function is a constant function.

Step-by-step solution

Understanding the Statement

To begin, it's important to grasp what the problem statement is asserting. In simpler terms, it is saying: If a function's slope (which is measured by the derivative) is zero at every point in its domain, then the function must be a constant function, which means its value does not change over its domain.

Exploring a Counter Example

In order to either prove or disprove this statement, consider a well-known function such as \(g(x)=|x|\). The derivative of this function at zero is undefined, while it is \(1\) for \(x>0\) and \(-1\) for \(x<0\). This function is clearly not constant, yet its derivative is \(0\) at some points.

The True Rule

The counter-example shows that the initial statement might not be completely accurate. However, the 'constant rate of change theorem' in calculus theorem states that if \`f\` is a differentiable real-valued function on an interval \([a, b]\) and \(f^{\prime}(x) = 0\) for all \(x\) in (a, b), then \`f\` is constant on \([a, b]\). In the given statement, it's assumed that the interval is the entire domain, not a specific interval, which makes it false.