Problem 55 - Calculus of a Single Variable

Question

The function \(f(x)=\left\\{\begin{array}{ll}{0,} & {x=0} \\ {1-x,} & {0< x \leq 1}\end{array}\right.\) is differentiable on \((0,1)\) and satisfies \(f(0)=f(1)\) However, its derivative is never zero on \((0,1) .\) Does this contradict Rolle's Theorem? Explain.

Step-by-Step Solution

Answer

No, this does not contradict Rolle's Theorem. Rolle's Theorem also requires the function to be continuous on the closed interval, which in this case, it is not.

Step 1: Identify the given function and its domain

The function \( f(x) \) is provided as \( f(x) = 0 \) when \( x = 0 \) and \( f(x) = 1 - x \) when \( 0 < x \leq 1 \). The function is differentiable on the interval \( (0, 1) \) and it is given that \( f(0) = f(1) \). However, it is stated that its derivative is never zero on this interval.

Step 2: Evaluate the function at the endpoints

Let's evaluate the function at the endpoints of the interval, i.e., at \( x = 0 \) and \( x = 1 \). By the definition of the function, we have \( f(0) = 0 \) and \( f(1) = 1 - 1 = 0 \))

Step 3: Check the conditions of Rolle's Theorem

With \( f(0) = f(1) \), the function does satisfy the condition of Rolle's Theorem that the function values at the endpoints of the interval should be equal. The function is also differentiable on the interval \( (0, 1) \). However, the condition that the function should be continuous on the closed interval \( [0, 1] \) is not mentioned.

Step 4: Analyse discrepancy with Rolle's Theorem

As per Rolle's Theorem, there should exist at least one point \( c \) in \( (0, 1) \) at which the derivative \( f'(c) = 0 \). But it is explicitly stated in the question that the derivative is nowhere zero in \( (0, 1) \). This seems to contradict Rolle's Theorem.

Step 5: Identify the missing condition

The problem here is that although the function is differentiable on \( (0, 1) \) and \( f(0) = f(1) \), it is not continuous on \( [0, 1] \), especially at \( x=0 \). The left-hand limit at \( x=0 \) is not equal to the right-hand limit. This makes the function discontinuous at \( x=0 \). Thus, one of the prerequisites for Rolle's Theorem does not hold, and this doesn't contradict Rolle's Theorem.

Key Concepts

Differentiability

Differentiability is a mathematical property that describes whether a function has a derivative at all points in its domain. If a function is differentiable at a point, it means that there exists a tangent line to the curve of the function at that point. This implies that the function is smooth in the neighborhood of that point, without any sharp corners or breaks.
In our exercise, the function is differentiable on the open interval \(0, 1\). This means that within this interval, the function smoothly transitions from one point to another. While it is differentiable in \(0, 1\), the overall continuity and differentiability over a closed interval \[0, 1\] must be interpreted considering endpoint behavior.

Continuity

Continuity is another property of functions that indicates the unbroken nature of their graph. A function is continuous over an interval if it can be drawn without lifting the pencil from the paper at every point within that interval. More formally, a function \(f(x)\) is continuous at a point \(x = a\) if the limit as \(x\) approaches \(a\) exists and equals \(f(a)\).
In the provided exercise, while the function \(f(x)\) is differentiable on \(0, 1\), it lacks continuity over the closed interval \[0, 1\]. Particularly at \(x=0\), \(f(x)\) jumps from \(1 - 0\) to \(0\). This points to a discontinuity at \(x=0\), which violates one of the prerequisites of Rolle's Theorem.

Endpoints Evaluation

Endpoints evaluation involves assessing the behavior of a function precisely at the starting and ending points of an interval. This is crucial for understanding properties like continuity and for satisfying certain theorems such as Rolle's Theorem.
In this problem, evaluating the function at \(x = 0\) gives \(f(0) = 0\), and evaluating at \(x = 1\) yields \(f(1) = 0\). This indicates that the function values at both endpoints are equal. However, because of the jump in values at \(x=0\) when moving from the left, the function does not meet the complete conditions needed by Rolle's Theorem.

Closed Interval

A closed interval \[a, b\] in mathematics includes all the points between \(a\) and \(b\) as well as the endpoints themselves. It is a vital consideration when applying theorems that require certain conditions to be met throughout the entire range of inputs.
Rolle's Theorem, for instance, specifically demands continuity on the closed interval \[a, b\] and differentiability on the open interval \(a, b\). In our scenario, although \(f(x)\) is differentiable on \(0, 1\), it lacks continuity on \[0, 1\] due to a jump at \(x=0\). This break in continuity over the closed interval invalidates the application of Rolle's Theorem.