Problem 46 - Calculus for Biology and Medicine

Question

Assume that \(f\) is continuous on \([a, b]\) and differentiable on \((a, b)\). Show that if \(f(a)

Step-by-Step Solution

Answer

By the Mean Value Theorem, if \( f(a) < f(b) \), there is a \( c \in (a, b) \) such that \( f'(c) > 0 \).

Step 1: Understand the Mean Value Theorem (MVT)

The Mean Value Theorem states that if a function \( f \) is continuous on \([a, b]\) and differentiable on \((a, b)\), then there exists at least one point \( c \) in \( (a, b) \) such that \( f'(c) = \frac{f(b) - f(a)}{b - a} \). This sets the groundwork for finding a point where the derivative is positive.

Step 2: Apply the Conditions of MVT

We know that \( f(a) < f(b) \). Thus, \( f(b) - f(a) > 0 \). Since \( b - a > 0 \) (as \( a eq b \)), the expression \( \frac{f(b) - f(a)}{b - a} \) is positive. Therefore, by MVT, there is at least one \( c \) in \( (a, b) \) where \( f'(c) > 0 \).

Step 3: Conclusion from MVT Application

By applying the Mean Value Theorem to the given conditions and the fact that \( f(a) < f(b) \), we conclude that \( f' \) is positive at some point between \( a \) and \( b \). This means there is a point \( c \) where the slope of the tangent (the derivative) is positive.

Key Concepts

Continuity

In mathematics, **continuity** is a property of a function where small changes in the input result in small changes in the output. Imagine a graph that you can draw without lifting your pencil off the paper; that's a continuous function. For the Mean Value Theorem to apply, the function must be continuous over the closed interval \( [a, b]\). This ensures there are no breaks or jumps in the graph on this interval.
Here's why that's crucial:

• It allows the function to smoothly transition between points \(a\) and \(b\).
• Ensures that every value between \(f(a)\) and \(f(b)\) is hit by the function.

Continuity is like a bridge that seamlessly connects two cities without interruption. It's this smoothness that lets us use calculus effectively to analyze the behavior of functions.

Differentiability

**Differentiability** takes continuity a step further by considering if the derivative exists at every point in an interval. For a function to be differentiable on an open interval \( (a, b)\), its graph must be smooth without any sharp corners or cusps within this range. Why does differentiability matter?
Because:

• It guarantees that the derivative, or the slope of the tangent, can be calculated accurately at any point \(c\) within \((a, b)\).
• It ensures the transition between points is smooth enough to apply the Mean Value Theorem.

Imagine skating on a perfect ice rink; the smoother the rink (differentiability), the better you can glide from one end to the other.

Derivative

The **derivative** of a function at a given point tells us the slope of the tangent to the function's graph at that point. In practical terms, it's like asking, "How steep is my hill at \(c\)?" For the Mean Value Theorem, we're interested in finding such a point \(c\) where the derivative meets certain conditions.
Specifically:

• If \(f(a) < f(b)\), then somewhere between \(a\) and \(b\), the derivative \(f'(c)\) must be positive.
• This means the slope at \(c\) is uphill, indicating a positive change over the interval.

Understanding derivatives is crucial for interpreting how the function behaves between any two points and predicting where a function might be increasing or decreasing.

Interval

An **interval** refers to the range of inputs under consideration, denoted here as \([a, b]\) and \((a, b)\). Knowing the type of interval helps in determining the applicable rules for analysis.

• \([a, b]\) is a closed interval, including both endpoints \(a\) and \(b\).
• \((a, b)\) is open, excluding these endpoints, tending to focus on the interior points.

In the context of the Mean Value Theorem, we require the function to be continuous on the closed interval \([a, b]\) and differentiable on the open interval \((a, b)\). This distinction is pivotal, as it ensures proper conditions for applying calculus techniques. It's like ensuring you're only looking at the inside of a sandwich, not worried about the crust edges when applying specific theorems.