Problem 18 - Calculus for Biology and Medicine

Question

[This problem illustrates the fact that \(f^{\prime}(c)=0\) is not a sufficient condition for the existence of a local extremum of a differentiable function.] Show that the function \(f(x)=x^{3}\) has a horizontal tangent at \(x=0 ;\) that is, show that \(f^{\prime}(0)=0\), but \(f^{\prime}(x)\) does not change sign at \(x=0\) and, hence, \(f(x)\) does not have a local extremum at \(x=0\).

Step-by-Step Solution

Answer

The derivative \( f'(0) = 0 \), showing a horizontal tangent, but \( f'(x) = 3x^2 \) does not change sign at \( x = 0 \), so no local extremum exists.

Step 1: Differentiate the Function

First, we need to differentiate the function \( f(x) = x^3 \). Using the power rule for differentiation, the derivative of \( f(x) \) is \( f'(x) = 3x^2 \).

Step 2: Evaluate the Derivative at x=0

Next, evaluate the derivative at the point \( x = 0 \). Substitute \( 0 \) into the derivative to find \( f'(0) = 3(0)^2 \). Thus, \( f'(0) = 0 \), indicating that there is a horizontal tangent at \( x = 0 \).

Step 3: Analyze the Sign of the Derivative Around x=0

Examine the sign of \( f'(x) = 3x^2 \) around \( x = 0 \). Notice that \( 3x^2 \) is always non-negative for all real \( x \), as squares of real numbers are always zero or positive. Thus, \( f'(x) \) is always greater than or equal to zero and never changes sign.

Step 4: Conclusion About Local Extremum

Because \( f'(x) \) does not change sign around \( x = 0 \), the function \( f(x) = x^3 \) does not have a local extremum at \( x = 0 \). The function is simply tangent to the horizontal line, but there is no maximum or minimum at this point.

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that involves finding how a function changes as its input changes. It essentially measures the function's rate of change or its slope at any given point. If you imagine a curve on a graph, differentiation tells you how steep the curve is at specific points.
To differentiate a function, you often rely on rules like the power rule to help make the process quick and straightforward.

• The result of differentiation is called the derivative.
• The derivative provides a precise way to describe the slope at any point of a function.
• In mathematical notation, the derivative of a function is often denoted as \( f'(x) \).

Through differentiation, we can analyze functions more deeply and observe their behavior, such as when they increase or decrease.

Extremum

In calculus, an extremum refers to either a maximum or minimum value that a function reaches. It represents points where the function takes on these extreme values. Extrema are important as they indicate where a function's behavior changes most significantly.
An extremum can be categorized into two types:

• Local Extremum: This is where the function has a maximum or minimum value within a small range around a point.
• Global Extremum: This occurs when the maximum or minimum is the absolute highest or lowest value over the entire range of the function.

Extrema are often found by identifying where the derivative of the function is zero, suggesting the slope of the tangent line is flat. However, just because the derivative is zero doesn't always ensure an extremum occurs, as seen with horizontal tangents.

Horizontal Tangent

A horizontal tangent occurs when the tangent line to a curve at a given point is flat, meaning its slope is zero. This happens when the derivative of the function equals zero at that point.
A horizontal tangent often suggests an extremum might occur, but it does not guarantee it. It's essential to check if the derivative changes sign around this point. If it changes from positive to negative or vice versa, then an extremum is present. If the derivative does not change sign, the horizontal tangent is simply a flat spot on the curve without a peak or valley.

• To examine this, you can observe the sign of the derivative before and after the point of interest.
• If the derivative remains the same sign, there's no minimum or maximum, just a horizontal tangent.

Understanding horizontal tangents helps in analyzing the behavior and structure of functions more comprehensively.

Power Rule

The power rule is one of the key rules for differentiation used to find the derivative of a function quickly and easily. It is specifically helpful when dealing with polynomial functions.
The power rule states that if you have a function of the form \( f(x) = x^n \), the derivative is given by \( f'(x) = nx^{n-1} \). This rule simplifies the process significantly and makes differentiating powers of \( x \) straightforward.

• For example, differentiating \( x^3 \) using the power rule results in \( f'(x) = 3x^2 \).
• This formula decreases the power by one and multiplies the original power to the coefficient.
• It's a fundamental tool in calculus, easing complex differentiation into manageable steps.

Mastering the power rule is essential for anyone learning calculus, as it forms the basis for tackling more complex problems.