Question

Explain why Rolle's Theorem cannot be applied to the function \(f(x)=|x|\) on the interval \([-a, a]\) for any \(a>0\).

Short answer

Based on the step-by-step solution above: Question: Explain why Rolle's Theorem cannot be applied to the function \(f(x) = |x|\) on the interval \([-a, a]\) for any \(a>0\). Answer: Rolle's Theorem requires a function to be continuous on a closed interval, differentiable on the open interval, and have the same end values. While the function \(f(x) = |x|\) is continuous on the closed interval \([-a, a]\) and has the same end values, it is not differentiable at \(x=0\) because its left and right derivatives are not equal. Therefore, Rolle's Theorem cannot be applied to this function on the interval \([-a, a]\) for any \(a>0\).

Step-by-step solution

Continuous on the closed interval \([-a, a]\)

For \(f(x) = |x|\), we have two cases: 1. If \(x\ge0\), then \(f(x) = x\). 2. If \(x<0\), then \(f(x) = -x\). In both cases, the function is continuous over the interval \([-a, a]\), as the two parts are polynomials. At the point \(x=0\), the left and right limits also exist and are equal. Hence, the function is continuous over the closed interval \([-a, a]\).

End values are the same

We need to check whether \(f(-a)=f(a)\). Since \(-a<0\), \(f(-a)=-(-a)=a\). Since \(a\ge0\), \(f(a)=a\). So \(f(-a)=f(a)\) as required.

Differentiable on the open interval \((-a, a)\)

Now, consider the function f(x) = |x| as two cases: 1. If \(x>0\), then \(f(x) = x\) and \(f'(x) = 1\). 2. If \(x<0\), then \(f(x) = -x\) and \(f'(x) = -1\). For \(x=0\), we need to check if both the left and right derivatives exist and are equal. 1. The right derivative (x>0) is: \(f'(0^{+}) = 1\). 2. The left derivative (x<0) is: \(f'(0^{-}) = -1\). Since \(f'(0^{+}) \ne f'(0^{-})\), we conclude that the function is not differentiable at x=0.

Conclusion

As the function \(f(x) = |x|\) is not differentiable on the open interval \((-a, a)\), Rolle's Theorem cannot be applied to this function on the interval \([-a, a]\) for any \(a>0\).

Key concepts

Absolute Value Function

The absolute value function is a fascinating concept in mathematics because it perfectly captures the non-negative distance between any real number and zero. If you take any real number, the absolute value will strip away any negative sign, effectively making all results non-negative. This behavior is essential for dealing with real-world quantities like distance, where direction does not matter.

In mathematical terms, the absolute value function is defined as follows:

• If the input, say \(x\), is positive or zero, the absolute value function simply returns \(x\).
• If \(x\) is negative, it returns \(-x\), effectively flipping the sign to make it positive.

This piecewise definition divides the absolute value function into two distinct parts, depending on whether the input is positive or negative. This nature classifies it under what we call piecewise functions.

Continuity and Differentiability

Continuity deals with the smoothness and unbroken nature of a function's graph. For a function to be continuous over an interval, there should be no abrupt jumps, breaks, or holes in its graph. The absolute value function \(f(x) = |x|\) is continuous over any interval, particularly on the closed interval \([-a, a]\). This is because its output smoothly transitions on any input \(x\), whether positive or negative.

However, differentiability is more stringent than continuity, as it requires the function to have a defined slope at each point in the interval. For a function to be differentiable, the derivatives from the left and right must equal at all points within the interval. This ensures that the graph has no sharp turns or corners.

• While the absolute value function is continuous, it is not differentiable at \(x = 0\). This point has a corner, where the slope of the graph from the left does not equal the slope from the right.

Because of this lack of differentiability at zero, certain theorems, specifically Rolle's Theorem, cannot be applied, as differentiability over the entire interval is required.

Piecewise Functions

Piecewise functions are wonderful tools in mathematics for modeling situations with multiple scenarios or rules. Essentially, these are defined by different expressions depending on the input value. They allow us to 'piece together' various functional behaviors into one overarching function.

The absolute value function is a classic example of a piecewise function. It is expressed as:

• \(f(x) = x\) if \(x \geq 0\)
• \(f(x) = -x\) if \(x < 0\)

This piecewise nature allows the absolute value function to reflect the numerical value without regard to the sign.

Piecewise functions offer flexibility but also exhibit points where continuity and differentiability might need thorough examination. It's these points, often where the function 'switches' from one piece to another, that can pose challenges similar to the differentiability issue at zero for the absolute value function. Understanding how these works can demystify several complex mathematical and real-world scenarios.