Problem 106 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Let $$ f(x)=\left\\{\begin{array}{ll} -3 x & \text { if } x<0 \\ 2 x+4 & \text { if } x \geq 0 \end{array}\right. $$ a. Compute \(f^{\prime}(x)\) and show that it changes sign from negative to positive as we move across \(x=0\). b. Show that \(f\) does not have a relative minimum at \(x=0 .\) Does this contradict the first derivative test? Explain your answer.

Step-by-Step Solution

Answer

The derivative of the function \(f(x)\) is given by \(f'(x) = \begin{cases} -3 & \text{if } x < 0 \\ 2 & \text{if } x \geq 0 \end{cases}\). It changes sign from negative to positive as we move across \(x=0\). However, there is no relative minimum at \(x=0\) since the function is discontinuous at that point, and the first derivative test is not applicable there. This does not contradict the first derivative test.

Step 1: Compute the derivative of the function

To compute the derivative of the function \(f(x)\), we need to differentiate each piece of the piecewise function separately (keep in mind that the function is defined differently for \(x<0\) and \(x \geq 0\)). So, we get: \(f'(x) = \begin{cases} -3 & \text{if } x < 0 \\ 2 & \text{if } x \geq 0 \end{cases} \)

Step 2: Analyze the derivative's behavior as x approaches 0

Let's check the value of \(f'(x)\) when \(x < 0\) and \(x\geq 0 \). When \( x < 0 \), we have \( f'(x) = -3 \), which is negative. When \( x \geq 0 \), we have \( f'(x) = 2 \), which is positive. Thus, the derivative \(f'(x)\) changes sign from negative to positive as we move across \(x=0\).

Step 3: Determine the existence of a relative minimum at x=0

According to the first derivative test, if the derivative of a function changes sign from negative to positive at a point, then the function has a relative minimum at that point. BHowever, in this case, the function \(f(x)\) is not continuous at \(x=0\) (as \(f(0^-) = -3 \times 0 = 0\) and \(f(0^+) = 2 \times 0 + 4 = 4\)). As a result, the first derivative test is not applicable here. Thus, even though the derivative changes sign from negative to positive as we cross \(x=0\), it doesn't imply that there is a relative minimum at \(x=0\) due to the discontinuity, and so it doesn't contradict the first derivative test.

Key Concepts

First Derivative Test

The first derivative test is a method used to identify potential relative extrema, such as minimum or maximum points of a function. It relies on examining the sign changes of the derivative. To apply the test, we first differentiate the function.

• If the first derivative changes from negative to positive at a certain point, the function has a relative minimum at that point.
• Conversely, if the derivative changes from positive to negative, the function has a relative maximum.

When analyzing piecewise functions, it is important to compute the derivative for each segment separately. For example, in the original exercise, the derivative changes sign across the point of interest, hinting at a relative minimum. However, this indication must be validated by checking the continuity of the function at that point.

Discontinuity

Discontinuity occurs when a function has a break, hole, or jump at certain points, meaning it does not draw a continuous line or curve. This can complicate analyses involving derivatives and relative extrema.
In a piecewise function, discontinuity often arises when switching from one prescribed function to another. At these points, the limit from the left does not equal the limit from the right, which affects the application of the first derivative test. In the given exercise, the function changes values at the point where the two rules switch, and the function is not continuous at \(x=0\) as the value from the left does not equal the value from the right.

Ensuring that a function is continuous is critical before performing tests like the first derivative test because the behavior of the function across the point needs to be smooth to make a reliable conclusion about any extreme values.

Relative Minimum

A relative minimum in a function is a point where the function has the lowest value in comparison to surrounding points. For smooth functions, this is typically determined by looking for where the derivative goes from negative to positive.
In the context of piecewise functions, finding relative minima requires special attention to each piece of the function, especially around points where the pieces connect as these can hide discontinuities. In the exercise, although the sign of the derivative changes, suggesting a relative minimum at \(x=0\), the discontinuity overrides this suggestion, showing no actual relative minimum at this point.

It's important to always consider both the derivative sign change and continuity at a point to correctly identify a relative minimum.

Differentiation

Differentiation is the process of finding the derivative of a function, which tells us the rate of change of the function's value with respect to its input. It is fundamental in calculus and pivotal in finding extrema points of functions.

• When facing a piecewise function, differentiation requires extra care. Each segment of the piecewise function is differentiated individually.
• This localized differentiation approach helps in understanding the behavior of the function in each of its segments.

In the original exercise, differentiation was conducted separately for the intervals determined by the piecewise function definition. The derivative tells us how the function changes within each segment, showing a negative rate of change before \(x=0\) and a positive one after \(x=0\).

Careful differentiation ensures that you analyze the true behavior of complex functions, helping in understanding potential points of interest like relative maxima or minima.