Problem 17 - 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 function \( f(x) = x^3 \) has a horizontal tangent at \( x=0 \), but no local extremum because \( f'(x) = 3x^2 \) does not change sign around \( x=0 \).

Step 1: Differentiate the Function

First, we need to find the derivative of the function \( f(x) = x^3 \). The derivative of \( x^3 \) is calculated using the power rule: \( \frac{d}{dx}(x^n) = n \cdot x^{n-1} \). For \( n = 3 \), we apply this rule to get \( f'(x) = 3x^2 \).

Step 2: Evaluate the Derivative at the Point

Next, we evaluate the derivative \( f'(x) \) at \( x = 0 \). Substitute \( x = 0 \) into \( f'(x) = 3x^2 \) to get: \( f'(0) = 3(0)^2 = 0 \). This shows the function has a horizontal tangent at \( x = 0 \).

Step 3: Check for Change of Sign in the Derivative

To determine if there is a local extremum at \( x = 0 \), we need to check if \( f'(x) \) changes sign around \( x = 0 \). Since \( f'(x) = 3x^2 \) is always non-negative (it equals zero or is positive for all \( x \)), the derivative does not change sign around \( x = 0 \).

Step 4: Conclude No Local Extremum Exists

Since \( f'(x) \) does not change sign around \( x = 0 \), \( f(x) = x^3 \) does not have a local extremum at this point despite having a horizontal tangent \( f'(0) = 0 \). A local extremum requires the derivative to be positive before the point and negative after, or vice versa.

Key Concepts

Differentiable Function

A differentiable function refers to a function that has a derivative at each point in its domain. This means the function is smooth and continuous, without any sudden jumps or sharp corners. In simple terms, you can draw it nicely on paper without lifting your pencil.
For a function to be differentiable, it must first be continuous. However, not every continuous function is differentiable. Differentiable functions allow us to use calculus tools to explore changes in behavior, such as finding slopes, determining extrema, and analyzing shape.
Examples of differentiable functions include polynomials, trigonometric functions, and exponential functions. In our exercise, the function we considered was a polynomial, specifically the cubic function, which is always differentiable.

Derivative

The derivative is a fundamental concept in calculus that measures how a function changes as its input changes. It is often thought of as the "slope" of the function at any given point. The derivative tells us how steep the graph of the function is at a particular point.
A derivative can be expressed as the limit:\[ f'(x) = \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h}\]
In the context of the exercise, we used the power rule to derive the function \( f(x) = x^3 \). This gives us \( f'(x) = 3x^2 \), indicating how the slope of the curve changes along different values of \( x \).

• For \( x = 0 \), the derivative is \( f'(0) = 0 \), showing a horizontal tangent.
• For any other \( x \), the derivative is positive, meaning the function is increasing.

Local Extremum

A local extremum refers to a point on a function where it reaches a local maximum or minimum. This is where a function changes direction from increasing to decreasing or vice versa.
Mathematically, a local extremum requires that the derivative, \( f'(x) \), be zero and change sign at the point. This tells us that the graph has a peak or a valley.
In our exercise, although the derivative at \( x = 0 \) is zero, \( f'(x) = 3x^2 \) does not change sign around zero. It remains non-negative, which means there's no peak or valley formed by the graph at \( x = 0 \). Thus, despite having a horizontal tangent, \( f(x) = x^3 \) does not have a local extremum at this point.

Horizontal Tangent

A horizontal tangent occurs when the derivative of a function at a specific point is zero. This suggests that the slope of the tangent line to the graph at that point is flat or horizontal.
Finding a horizontal tangent involves setting the derivative equal to zero. In our exercise, this was demonstrated at \( x = 0 \) where the derivative \( f'(x) = 3x^2 \) evaluated to zero, indicating a horizontal tangent.
However, it's crucial to remember that a horizontal tangent doesn't always signify a local extremum. As shown, the derivative \( f'(x) \) should also change sign at the point to confirm a local extremum. In our case, since no sign change occurs around \( x = 0 \), no local extremum is present.