Problem 19 - University Calculus: Early Transcendentals

Question

Show that if \(f^{\prime \prime} \geq 0\) throughout an interval \([a, b],\) then \(f^{\prime}\) has at most one zero in \([a, b] .\) What if \(f^{\prime \prime}<0\) throughout \([a, b]\) instead?

Step-by-Step Solution

Answer

If \(f'' \geq 0\), \(f'\) has at most one zero. If \(f'' < 0\), \(f'\) also has at most one zero.

Step 1: Understand the Given Information

We are given that the second derivative of a function, \(f''\), is greater than or equal to zero throughout the interval \([a, b]\). This means \(f''\) is non-negative, indicating that the function \(f(x)\) is convex over this interval.

Step 2: Implications of Non-Negative Second Derivative

Since \(f'' \geq 0\) over \([a, b]\), the first derivative \(f'\) is a non-decreasing function. In simpler terms, \(f'\) cannot decrease as \(x\) moves from \(a\) to \(b\).

Step 3: Analyzing the Zeros of the First Derivative

For \(f'\) to have more than one zero, it would need to go from positive to zero and back to positive, or from negative to zero and back to negative. However, since \(f'\) is non-decreasing, once \(f'\) reaches zero, it cannot become negative again over the interval. Thus, it can have at most one zero.

Step 4: Verify for \(f'' < 0\) Case

If \(f'' < 0\) throughout \([a, b]\), then \(f'\) is a strictly decreasing function, meaning it always goes down as \(x\) moves from \(a\) to \(b\). In this case, \(f'\) can still have at most one zero, as it would cross zero only once, transitioning from positive to negative.

Key Concepts

Convex Function

A function is considered convex over an interval when a line segment between any two points on the function's graph does not fall below the graph itself. This implies that for any two points

• If the second derivative, denoted as \(f''(x)\), is greater than or equal to zero, the function is non-decreasing or constant.
• Graphically, the curve opens upwards, like a bowl, which indicates that any tangent to any point on the function lies below the graph or just touches it.

Convex functions have a few interesting properties, such as:

• Their global minimum, if it exists, is at the lowest point of the curve within the interval.
• They play a crucial role in optimization problems because they help in determining points where functions reach their minimum value.

Understanding convexity helps us analyze characteristics of functions as they relate to their graph's shape and derivative behavior. In mathematical disciplines like calculus, this concept provides foundational insights for examining function behavior over defined intervals.

Derivative

Derivatives represent the rate of change of a function with respect to a variable. In simple terms, a derivative can tell you how fast or slow something changes. For a given function \(f(x)\):

• The first derivative, denoted as \(f'(x)\), provides insights into the slope of the function or how steep the curve is at a particular point.
• When \(f'(x) > 0\), the function is increasing.
• When \(f'(x) < 0\), the function is decreasing.
• When \(f'(x) = 0\), the function may be at a local maximum, minimum, or a point of inflection.

Derivatives help us analyze important characteristics of functions, especially when solving problems related to motion, optimization, and modeling. Understanding the derivative helps students to comprehend concepts like slope, instantaneous rate of change, and tangents, which are vital in calculus.

Second Derivative

The second derivative, noted as \(f''(x)\), delves deeper into the behavior of the original function by examining the derivative of \(f'(x)\). This can be imagined as analyzing the rate of change of the rate of change. The second derivative offers critical insights into a function's concavity and the nature of its turning points:

• A positive second derivative \(f''(x) > 0\) over an interval indicates a convex function, meaning a curve that bends upwards.
• A negative second derivative \(f''(x) < 0\) signifies a concave function, where the curve bends downwards, similar to an upside-down bowl.
• If the second derivative is zero, it's necessary to examine further to conclude about concavity at that point.

In the original exercise, determining whether \(f'(x)\) has zeroes relies heavily on understanding \(f''(x)\). The second derivative provides crucial information, with a positive \(f''(x)\) rendering the derivation non-decreasing so that \(f'(x)\) cannot have more than one zero within the given interval. This knowledge is vital for assessing function behavior, especially when distinguishing between increasing and decreasing function segments.