Problem 47 - Calculus for Biology and Medicine

Question

Assume that \(f\) is continuous on \([a, b]\) and differentiable on \((a, b)\). Assume further that \(f(a)=f(b)=0\) but \(f\) is not constant on \([a, b] .\) Explain why there must be a point \(c_{1} \in(a, b)\) with \(f^{\prime}\left(c_{1}\right)>0\) and a point \(c_{2} \in(a, b)\) with \(f^{\prime}\left(c_{2}\right)<0 .\)

Step-by-Step Solution

Answer

There are points \(c_1\) where \(f'(c_1) > 0\) and \(c_2\) where \(f'(c_2) < 0\) due to \(f\) increasing/decreasing around local extrema within \((a, b)\).

Step 1: Understanding the conditions

We are given a continuous function \(f\) on the closed interval \([a, b]\) and differentiable on the open interval \((a, b)\). We also know that \(f(a) = f(b) = 0\) and that \(f\) is not constant on \([a, b]\). Our goal is to deduce the existence of points \(c_1\) and \(c_2\) in \((a, b)\) where \(f'\left(c_{1}\right)>0\) and \(f'\left(c_{2}\right)<0\).

Step 2: Utilize the maximum and minimum values

Since the function is not constant and is continuous on \([a, b]\), it must achieve a maximum or minimum value at some point in \((a, b)\) due to the Extreme Value Theorem. These local extrema must be within \((a, b)\) because \(f(a) = f(b) = 0\) and \(f\) is not constant.

Step 3: Apply the First Derivative Test

At any local maximum or local minimum within an open interval where the function is differentiable, the derivative \(f'\) must be equal to zero. However, since we need points with positive or negative derivatives, we must consider the behavior around these extrema.

Step 4: Identify intervals of increase or decrease

Given that \(f\) achieves a non-zero maximum or minimum between \(a\) and \(b\), \(f\) must have intervals where it is either increasing to reach the maximum or decreasing to reach the minimum. If \(f\) increases, \(f' > 0\); if it decreases, \(f' < 0\).

Step 5: Conclusion using Mean Value Theorem and Rolle’s Theorem

By the Mean Value Theorem, since \(f(a) = f(b) = 0\) but \(f\) is not constant, there must be points \(c_1\) and \(c_2\) where \(f'\left(c_{1}\right) > 0\) and \(f'\left(c_{2}\right) < 0\). These are achieved in intervals of increase or decrease between the extrema of the function.

Key Concepts

Rolle's Theorem

Rolle's Theorem is a fundamental concept in calculus that comes with a beautiful geometric aspect. It is a special case of the Mean Value Theorem, which provides conditions under which a continuous and differentiable function must have a horizontal tangent line. Essentially, Rolle's Theorem states that if a function \(f\) is continuous on a closed interval \([a, b]\), differentiable on the open interval \((a, b)\), and verifies \(f(a) = f(b)\), then there exists some point \(c\) in the interval \((a, b)\) such that \(f'(c)=0\). In simpler terms, there's at least one point where the slope of the tangent to the curve is flat (horizontal).

• The function must be continuous across the interval \([a, b]\).
• The function must be differentiable on the interval \((a, b)\).
• The function values at the endpoints must be the same, \(f(a) = f(b)\).

Understanding this theorem is vital because it helps identify the presence of peaks or troughs in the function's graph, even when the endpoints suggest a flat start and end.

Continuous Function

A continuous function is one that does not "jump" or have any abrupt changes in value. Imagine a smooth, unbroken line on a graph; that's how a continuous function behaves. Mathematically, for function \(f\) to be continuous on an interval \([a, b]\), it must be continuous at every point in that interval, meaning:

• For any point \(c\) within \([a, b]\), the limit as \(x\) approaches \(c\) must exist.
• \(f(c)\) must equal the limit as \(x\) approaches \(c\).

The fact that our function \(f\) in the problem is continuous on \([a, b]\) allows us to use the Extreme Value Theorem, which guarantees the function attains maximum and minimum points within the interval. This property is crucial for locating increases and decreases within the function, as changes must happen smoothly from one point to another.

Differentiable Function

A differentiable function is one that has a derivative at each point in its domain. This means the function has a defined slope or tangent line at every point, ensuring "smoothness" without sharp corners or cusps. For a function to be differentiable on an interval \((a, b)\):

• It must not have any breaks, asymptotes, or corners within the interval.
• The derivative \(f'\) must exist within the open interval \((a, b)\).

In our exercise, the differentiability of function \(f\) allows us to use the First Derivative Test, a tool for identifying local maxima and minima by examining the changes in sign of \(f'\). Differentiability combined with continuity is what makes many critical calculus theorems, including Mean Value Theorem and Rolle's Theorem, applicable.

First Derivative Test

The First Derivative Test is a handy tool in determining where a function is increasing or decreasing, and it's particularly useful for finding local maximum and minimum points. When you consider a differentiable function, here's how it works:

• A positive derivative, \(f'(c) > 0\), indicates the function is increasing at \(c\).
• A negative derivative, \(f'(c) < 0\), indicates the function is decreasing at \(c\).
• If \(f'(c) = 0\), \(c\) is a potential local extremum, which could be a peak or a trough, depending on the behavior on either side of \(c\).

In the context of our exercise, the First Derivative Test helps us reason why there must be points where the function is increasing or decreasing. Since \(f(a) = f(b) = 0\) and \(f\) is non-constant, it must reach a peak (increases, then decreases) and a trough (decreases, then increases) between \(a\) and \(b\). This is crucial for establishing the presence of \(c_1\) and \(c_2\) in the solution, aligning perfectly with the application of the Mean Value and Rolle's Theorems.