Problem 76 - Calculus of a Single Variable

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.

Step-by-Step Solution

Answer

The statement is true. A function with its derivative equal to zero at all points within its domain is indeed a constant function.

Step 1: Understanding the Statement

The first step is to understand the implications of the statement. The statement declares that if the derivative of a function is zero at all points in its domain, then that function is constant. This might suggest a direct relationship implying that the function doesn't change its value when the rate of change of the function is zero.

Step 2: Proving or Disproving the Statement

In order to prove or disprove this statement, one can use the fundamental concept of calculus that states: if a function is differentiable in an interval and its derivative is zero in that interval, it means that the function is constant in that interval. It is important to notice that the differentiability of a function implies its continuity.

Step 3: Justifying the Statement

The statement given is True. This is because the derivative of a function provides us the slope of the tangent line at any point in its domain. If this slope is always zero, it means the tangent is a horizontal line at all points, suggesting that the function doesn't change its value or in other words, the function is constant at all those points. So, if \(f^{\prime}(x)=0\) for all \(x\) in the domain of \(f,\) then \(f\) is a constant function. Hence, the statement is true.