Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the critical points and use the test of your choice to decide which critical points give a local maximum value and which give a local minimum value. What are these local maximum and minimum values? $$ f(t)=t-\frac{1}{t}, t \neq 0 $$

Step-by-Step Solution

Answer

The function has no critical points, hence no local max/min values.

Step 1: Find the First Derivative

To find the critical points, we first calculate the derivative of the function. The function is given by \( f(t) = t - \frac{1}{t} \). Differentiating with respect to \( t \), we get:\[ f'(t) = 1 + \frac{1}{t^2}.\]

Step 2: Set the First Derivative to Zero

Critical points occur where the derivative is zero or undefined. Set the derivative equal to zero to find:\[1 + \frac{1}{t^2} = 0.\]This equation has no real solutions since \( \frac{1}{t^2} = -1 \) is not possible for any real number \( t \). Thus, there are no critical points from setting the derivative to zero.

Step 3: Identify Where Derivative is Undefined

Since the critical points can also occur where the derivative is undefined, we look at the derivative \( f'(t) = 1 + \frac{1}{t^2} \). This derivative is undefined when \( t = 0 \). However, \( t = 0 \) is not in the domain of \( f(t) \) since \( t eq 0 \). Therefore, we need to analyze further the behavior for possible critical points.

Step 4: Determine Monotonicity

Since there are no critical points where \( f'(t) = 0 \), let's examine the behavior of \( f'(t) = 1 + \frac{1}{t^2} \). Notice that \( \frac{1}{t^2} \) is always non-negative, making \( 1 + \frac{1}{t^2} > 0 \) for all \( t eq 0 \). Therefore, \( f'(t) \) is always positive for \( t eq 0 \), indicating that \( f(t) \) is increasing over its entire domain.

Step 5: Conclude the Behavior of the Function

Since the function \( f(t) = t - \frac{1}{t} \) is always increasing where defined (i.e., for both \( t > 0 \) and \( t < 0 \)), there are no local maximum or minimum values. The function does not have turning points.

Key Concepts

Derivative of a Function

The first step in finding critical points is to determine the derivative of the function. A derivative, in simple terms, offers us the rate at which the function changes. For a function like \( f(t) = t - \frac{1}{t} \), differentiating with respect to \( t \), we use standard differentiation rules:

• The derivative of \( t \) is \( 1 \).
• The derivative of \( -\frac{1}{t} \) is \( \left(\frac{1}{t^2}\right) \). This comes from the power rule where \( x^{-n} \) becomes \(-nx^{-n-1}\).

So, the derivative of \( f(t) \) is \( f'(t) = 1 + \frac{1}{t^2} \). Finding the derivative is crucial because critical points often occur where this derivative equals zero or is undefined. In this instance, attempting to find \( f'(t) = 0 \) did not yield any real solutions. However, understanding this allows us to spot the tendencies of functions across their domains.

Monotonicity

Monotonicity refers to the increasing or decreasing nature of a function. Once we obtain the derivative, we can determine the monotonicity by examining its sign across the domain of the function. For \( f(t) = t - \frac{1}{t} \), the derivative \( f'(t) = 1 + \frac{1}{t^2} \) is always greater than zero for \( t eq 0 \).

This positive sign of the derivative indicates that the function is monotonically increasing wherever it's defined. Because there are no critical points from \( f'(t) = 0 \), the function does not switch directions. Hence, \( f(t) \) rises steadily for all \( t > 0 \) and \( t < 0 \), without any drops or turning points.

Local Maximum and Minimum

Local maximum and minimum points occur where a function changes direction; however, not all functions have these turning points. By setting the derivative to zero, we can find potential candidates for these points. Yet, as seen with \( f(t) = t - \frac{1}{t} \), sometimes the function lacks real solutions for zero slopes.

In this case, since \( f'(t) = 1 + \frac{1}{t^2} \) never equals zero, \( f(t) \) does not have local maxima or minima. It simply does not turn around. This situation shows how a function can be on a consistent upward path without the peaks and valleys that characteristically define local maxima or minima.

Function Analysis

Analyzing the behavior of a function involves looking at various properties, such as its derivatives, critical points, and overall direction. With \( f(t) = t - \frac{1}{t} \), we first identified its derivative and discovered that it has no point where it becomes zero, nor does it become undefined in the domain we consider.

Knowing the derivative's behavior, we concluded that the function is continuously increasing. There are no local maxima or minima, and the absence of critical points suggests this increasing function moves linearly along its path. For students, recognizing these characteristics offers a comprehensive way to predict function behavior and get an impression of its structure across its domain, thereby deepening their understanding of calculus and function properties.