Problem 17 - A Course in Calculus and Real Analysis

Question

Let \(f:[a, b] \rightarrow \mathbb{R}\) be continuous and consider the function \(F:[a, b] \rightarrow \mathbb{R}\) given by \(F(x):=\int_{a}^{x} f(t) d t\) for \(x \in[a, b]\). If \(f(x) \geq 0\) for all \(x \in[a, b]\), then show that \(F\) is monotonically increasing on \([a, b]\), and if \(f\) monotonically increasing on \([a, b]\), then \(F\) is convex on \([a, b]\). (Hint: Part (i) of Proposition \(4.27\) and Part (i) of Proposition 4.31.)

Step-by-Step Solution

Answer

To show that \(F(x) := \int_a^x f(t) dt\) is monotonically increasing on [a, b] when \(f(x) \geq 0\), we can use Proposition 4.31(i). Since \(f\) is continuous, \(F'(x) = f(x) \geq 0\) for all \(x \in [a, b]\), so by the proposition, \(F\) is monotonically increasing. To show that \(F\) is convex on [a, b] when \(f\) is monotonically increasing, we can also use Proposition 4.31(i). In this case, since \(f\) is monotonically increasing, \(f'(x) \geq 0\), so \(F''(x) = f'(x) \geq 0\) for all \(x \in [a, b]\), making \(F\) convex.

Step 1: Define the function F' and show it exists

Since \(f\) is continuous on \([a, b]\), the function \(F(x) := \int_a^x f(t) dt\) is differentiable and its derivative exists and is equal to \(f(x)\) for all \(x \in [a, b]\) by the First Fundamental Theorem of Calculus. So we have: \(F'(x) = f(x)\)

Step 2: Use Proposition 4.31(i) to show F is monotonically increasing

Given that \(f(x) \geq 0\) for all \(x \in [a, b]\), we know that \(F'(x) = f(x) \geq 0\). Now using Proposition 4.31(i), we can conclude that \(F\) is monotonically increasing on \([a, b]\). Part 2: Prove that \(F\) is convex on \([a, b]\) if \(f\) is monotonically increasing on \([a, b]\).

Step 1: Define the function F'' and show it exists

Since \(f\) is continuous on \([a, b]\), we have already established that \(F(x) := \int_a^x f(t) dt\) is differentiable. Now if \(f\) is monotonically increasing on \([a, b]\), then by the Second Fundamental Theorem of Calculus, the function \(F'(x)\) is also differentiable and its second derivative exists. We have: \(F''(x) = f'(x)\)

Step 2: Use Proposition 4.31(i) to show F is convex

We know that if a function has its second derivative greater than or equal to 0, then the function is convex. Since \(f(x)\) is monotonically increasing on \([a, b]\), we get \(f'(x) \geq 0\) for all \(x \in [a, b]\). Therefore, using our formula for the second derivative of \(F\), we have: \(F''(x) = f'(x) \geq 0\) Now using Proposition 4.31(i), we can again conclude that \(F\) is convex on \([a, b]\).

Key Concepts

Monotonic Functions

A function is called monotonic when it is consistently increasing or decreasing throughout its domain. In simpler terms, it does not change directions. When discussing the Fundamental Theorem of Calculus, consider the function \( F(x) = \int_a^x f(t) dt \) which represents the accumulation of area under the curve of \( f \) from \( a \) to \( x \).
If \( f(x) \geq 0 \) for all \( x \) in [a, b], then \( F(x) \) collects only non-negative amounts, meaning additional area is being added on as \( x \) increases. This implies that \( F(x) \) gets larger but never smaller as you move along the interval \([a, b]\), hence, \( F(x) \) is monotonically increasing.
By the First Fundamental Theorem of Calculus, the derivative \( F'(x) \) of such a function \( F \) would be \( f(x) \), leading us to conclude that since \( F'(x) = f(x) \geq 0 \), the function \( F \) does not decrease, affirming its monotonous nature.

Convexity

Convexity is a property of functions where every line segment drawn between any two points on the graph of the function lies above the graph itself. For \( F(x) \) as defined earlier to be convex on \([a, b]\), the second derivative \( F''(x) \) must be non-negative throughout the domain.
In our context, if \( f \) is monotonically increasing on \([a, b]\), then \( f'(x) \), the derivative of \( f \), is \( \geq 0 \). According to the Second Fundamental Theorem of Calculus, this derivative \( f'(x) \) translates into \( F''(x) = f'(x) \geq 0 \), meaning the slope of \( F'(x) \) is non-decreasing. This is indicative of the function \( F \) curving upwards, establishing its convex nature.
This concept is pivotal because it assures us that \( F \) maintains a stable, upward-sloping trend that doesn't dip or buckle, adhering to the criteria for a convex function.

Differentiability

Differentiability is the mathematical property indicating that a function has a derivative at all points in its domain. It's a strong sign of a function's smoothness. For the integral function \( F(x) = \int_a^x f(t) dt \), knowing \( f \) is continuous makes \( F \) differentiable, as asserted by the First Fundamental Theorem of Calculus.
This means that not only can we determine how \( F \) varies continuously, but we can also compute its rate of change at any point \( x \) in \([a, b]\) using the derivative \( F'(x) = f(x) \).
Further, when \( f(x) \) is monotonically increasing, the differentiability extends to the second derivative \( F''(x) \), establishing how quickly the rate of increase itself is changing. Simply put, differentiability assures us of the general smoothness of both the function and its rate of change, allowing us to confidently discuss concepts like monotonicity and convexity in a structured manner.