Problem 22 - Essential Calculus

Question

Use a graphing utility to graph the function on the closed interval [a,b]. Determine whether Rolle's Theorem can be applied to \(f\) on the interval and, if so, find all values of \(c\) in the open interval \((a, b)\) such that \(f^{\prime}(c)=0\). $$ f(x)=x-x^{1 / 3}, \quad[0,1] $$

Step-by-Step Solution

Answer

Yes, Rolle's Theorem can be applied to the function on the interval [0, 1] and the value of 'c' is 1/9 where the derivative of the function is zero.

Step 1: Graph the function

Use any graphing tool to graph the function \(f(x) = x - x^{1/3}\) on the interval [0, 1]. The continuous graph will help visualize whether the function meets the conditions of Rolle’s Theorem.

Step 2: Verify the conditions

Ensure the function meets the conditions for Rolle's Theorem. Here, \(f(x) = x - x^{1/3}\) is continuous and differentiable on the interval (0, 1) and \(f(0) = f(1) = 0\)

Step 3: Derive the function

Find the derivative of the function, \(f'(x) = 1 - (1/3) x^{-2/3}\).

Step 4: Find 'c'

This step involves finding the value of 'c' for which \(f'(c) = 0\). From the derivative, set it to zero and solve for 'c', where \(1 - (1/3) c^{-2/3} = 0\). Rearrange to get \(c^{-2/3} = 3/1\) or \(c = (1/3)^{-3/2} = 1/27^{1/2} = 1/3\^2 = 1/9\). So, \(c = 1/9\) is the value on the interval (0, 1) where the derivative of \(f(x)\) equals zero, satisfying Rolle's Theorem.

Key Concepts

Graphing Functions

Graphing functions is a fundamental skill in calculus and aids in visualizing the behavior of functions over a certain domain. To graph a function, one plots the outputs that correspond to various inputs, creating a visual representation of the relationship between the two.

For example, when graphing the function given in the exercise,  \(f(x) = x - x^{1/3}\) on the interval [0,1], using a graphing utility allows you to see how the function increases and where it may have flat points (where the slope of the tangent is zero). Visually assessing the graph can give us clues about where to look for certain features of a function, like the existence of horizontal tangents or intercepts with the axes.

Graphing is not only about getting the shape right but also understanding the function’s properties within the context of the given problem, such as confirming if a function is continuous and differentiable within a certain interval - both prerequisites for applying Rolle's Theorem.

Continuous Functions

Continuous functions are those that do not have any breaks, jumps, or holes. In layman’s terms, you can draw them without lifting your pencil from the paper. Formally, a function \(f(x)\) is continuous at a point \(x = a\) if, as the input values approach \(a\), the output of the function approaches \(f(a)\).

In the exercise, we're interested in whether the function \(f(x) = x - x^{1/3}\) is continuous over the closed interval [0,1]. The function's formula does not involve any terms that would cause division by zero or any other undefined operations within this interval. So, as per the solution, it meets one of the key conditions of Rolle's Theorem, as the function is continuous on the closed interval [0,1]. Understanding continuity is crucial because only continuous functions have derivatives at all points within their domains - another piece that links back to Rolle's Theorem.

Derivatives

The derivative of a function represents the rate at which a function's output changes as its input changes. In graphical terms, the derivative at any point on the function's graph is the slope of the tangent line to the graph at that point. Calculus students learn to find derivatives using symbolic methods and rules for differentiation.

For instance, the derivative of \(f(x) = x - x^{1/3}\) is found to be \(f'(x) = 1 - (1/3)x^{-2/3}\) by applying power rule for differentiation. The function and its derivative must be continuous and defined on the interval (0, 1) to satisfy Rolle's Theorem. In our case, the function is differentiable on the open interval (0, 1), which is the second condition checked in the solution provided for Rolle's Theorem.

Critical Points

Critical points of a function are where the derivative is zero or undefined. These points are often where the function has a local maximum or minimum, or a point of inflection. To locate them, one typically sets the derivative of the function equal to zero and solves for the corresponding input values.

In the example from our exercise, the critical point is found by setting \(f'(x) = 1 - (1/3) x^{-2/3} = 0\) and solving for \(x\), which gives us the value of \(x = 1/9\). As per Rolle's Theorem, since \(f(x)\) is continuous on [0,1] and differentiable on (0,1) with \(f(0) = f(1)\), there must be at least one point in the open interval (0, 1), where the derivative is zero. For our exercise, \(x = 1/9\) is that critical point which satisfies Rolle's Theorem.