Problem 81 - Calculus Single Variable

Question

Darboux's Theorem (after Jean Darboux, \(1842-1917\) ) states that if \(f^{\prime}\) exists at every point of an open interval containing \([a, b],\) and if \(\gamma\) is between \(f^{\prime}(a)\) and \(f^{\prime}(b),\) then there is a \(c\) in the interval \((a, b)\) such that \(f^{\prime}(c)=\gamma .\) The existence of such a \(c\) follows from the Intermediate Value Theorem if \(f^{\prime}\) is assumed continuous. Darboux's Theorem, which you will prove in this exercise, tells us that the assumption of continuity is not needed for functions that are derivatives. Suppose for definiteness that \(f^{\prime}(a)<\gamma<\) \(f^{\prime}(b) .\) Define \(g\) on \([a, b]\) by \(g(x)=f(x)-\gamma x\) a. Explain why \(g\) has a minimum value that occurs at some point \(c\) in \((a, b)\). b. Show that \(f^{\prime}(c)=\gamma\). c. Use Darboux's Theorem to show that if \(f\) increases on the interval \((l, c)\) and decreases on the interval \((c, r)\) (or vice versa), then \(f^{\prime}(c)=0\)

Step-by-Step Solution

Answer

The function \(g(x)\) has a minimum at some \(c\), and \(f'(c) = \gamma\). Darboux's Theorem confirms \(f'(c) = 0\) when \(f\) changes direction.

Step 1: Understanding Function g(x)

Define the function \[ g(x) = f(x) - \gamma x \] on \([a, b]\). The derivative of this function is \[ g'(x) = f'(x) - \gamma \]. We aim to show that there is a minimum at some point \(c\) in \((a, b)\).

Step 2: Applying the Extreme Value Theorem

Since \(g(x)\) is continuous on the closed interval \([a, b]\), by the Extreme Value Theorem, \(g\) has a minimum at some point in the interval. Furthermore, since \(a\) and \(b\) are within this interval, the minimum must occur at some point \(c\) in \((a, b)\) because \(g'(a) = f'(a) - \gamma < 0\) and \(g'(b) = f'(b) - \gamma > 0\).

Step 3: Showing \(f'(c) = \gamma\)

At the point where \(g(x)\) achieves its minimum, the derivative \(g'(c) = 0\). Thus, \(f'(c) - \gamma = 0\), which implies \[ f'(c) = \gamma \]. This shows the Intermediate Value Theorem for \(f'\) at \(c\).

Step 4: Using Darboux's Theorem for \(f'(c) = 0\)

Consider a function \(f\) such that it increases on \((l, c)\) and decreases on \((c, r)\), or vice versa. According to Darboux's Theorem applicable to \(f'(c)\), which does not need continuity, there must exist a point \(c\) where the derivative of \(f\) is zero (since the function changes from increasing to decreasing, or vice versa). Hence, \(f'(c) = 0\).

Key Concepts

Intermediate Value Theorem

The Intermediate Value Theorem (IVT) is a fundamental concept in calculus. It states that if a continuous function, say \(h\), takes two values \(h(a)\) and \(h(b)\) at either end of an interval \([a, b]\), then it must also take any value between \(h(a)\) and \(h(b)\) at some point within that interval.
This ensures that, within the interval \((a, b)\), any value between these outputs will have at least one corresponding input. For example, if \(h(a) = 2\) and \(h(b) = 6\), then for any value between 2 and 6, there exists some \(c\) in \((a, b)\) where \(h(c)\) equals that value.

• The function must be continuous on the closed interval \([a, b]\).
• The theorem primarily applies to real-valued functions.

This principle is vital for proving the existence of solutions and is instrumental in many mathematical proofs, including Darboux's Theorem, where a similar idea applies to derivatives.

Extreme Value Theorem

The Extreme Value Theorem (EVT) is another cornerstone in calculus. This theorem tells us that if a function \(g\) is continuous on a closed interval \([a, b]\), then \(g\) will attain a maximum and a minimum value within this interval.
The EVT assures us of two things:

• \(g\) has absolute maximum and minimum values within \([a, b]\).
• These extreme values occur either at the endpoints \(a\) or \(b\), or at some point \(c\) in the open interval \((a, b)\).

In the context of our exercise, the EVT was applied to a continuous function \(g(x) = f(x) - \gamma x\) over \([a, b]\), ensuring a minimum exists, which is crucial for showing specific properties of the function derivative.
However, the EVT holds primarily for continuous functions, reminding students that continuity is a key component for applying these theorems effectively.

Function Derivatives

In calculus, the derivative of a function \(f\), denoted \(f'(x)\), provides a central way of understanding how \(f\) changes. It represents the rate at which \(f\) changes with respect to its input variable.
The derivative can be thought of as the slope of the tangent line to the graph of \(f\) at a given point. It tells us:

• If \(f'(x) > 0\), the function is increasing at that point.
• If \(f'(x) < 0\), the function is decreasing.
• If \(f'(x) = 0\), there might be a maximum, minimum, or some kind of inflection point.

Derivatives are critical for optimization and in analyzing the behavior of functions. In the context of Darboux's Theorem, even if \(f'\) is not continuous, it still satisfies a property akin to the Intermediate Value Theorem for functions — highlighting its unique nature.
This perspective emphasizes how derivatives can be used regardless of continuity to find specific values, aligning with our exercise's explorations.

Piecewise Function Behavior

Piecewise functions are defined by different expressions over different parts of their domain. These types of functions can exhibit various behaviors depending on the interval being considered.
A piecewise function might increase in certain sections and decrease in others, leading to interesting properties at transition points. For example, at a transition point where a function changes from increasing to decreasing, there typically exists a point where the function derivative equals zero, implying a local extremum.
This idea plays into Darboux's Theorem, particularly when considering that even if the derivative \(f'\) is not continuous, it must satisfy certain behavior dictated by the larger function structure. For example:

• In \((l, c)\), the function might increase.
• In \((c, r)\), it can decrease.

Understanding these behaviors aids in visualizing the changes in the function and utilizing the theorem's implications to predict function behavior at specific points, such as the existence and nature of local mimina or maxima.