Problem 39 - Calculus of a Single Variable

Question

Applying the First Derivative Test In Exercises \(17-40\) , (a) find the critical numbers of \(f\) (if any), (b) find the open interval(s) on which the function is increasing or decreasing, (c) apply the First Derivative Test to identify all relative $$ f(x)=\left\\{\begin{array}{ll}{3 x+1,} & {x \leq 1} \\ {5-x^{2},} & {x>1}\end{array}\right. $$

Step-by-Step Solution

Answer

There are no critical numbers and hence cannot apply the First Derivative test. The function \(f(x)\) is increasing on the interval \(-\infty, 1]\) and decreasing on the interval \(1, \infty)\).

Step 1: Find the first derivative

First we need to find the derivative of \(f(x)\). This is done differently for each part of the piecewise function. For \(x \leq 1\), \(f'(x)=3\), and for \(x >1\), \(f'(x)=-2x\).

Step 2: Find critical numbers

Critical numbers are where the derivative of f is zero or undefined. Clearly, for \(x \leq 1\), derivative \(f'(x)=3\) does not have any critical number as it is linear and constant. For \(x >1\), solving \(f'(x)=-2x=0\) implies that \(x=0\); however, it won't be considered as it falls into first piecewise function's domain. Therefore, there are no critical numbers.

Step 3: Find intervals of increase or decrease

We can see that for \(x \leq 1\), the derivative \(f'(x)=3\) is always positive, so \(f(x)\) is increasing on the interval \((- \infty, 1]\). For \(x >1\), the derivative is negative, so \(f(x)\) is decreasing on the interval \((1, \infty)\).

Step 4: Apply the First Derivative Test

The First Derivative Test is a method to determine where a function has a local maximum or minimum. However with the function setup here, there are no critical points, hence we can't use the First Derivative Test in this case.

Key Concepts

Critical Numbers

When we study functions, critical numbers play a significant role in understanding the function's behavior. Critical numbers are the values of the independent variable, usually denoted as 'x', where the first derivative of the function is either zero or undefined. These are potential points where a function can have a peak or a trough, meaning they could be where the function attains a local maximum or minimum.

For example, with the function presented in the exercise, the first part of the function, defined for \( x \leq 1 \), is a straight line with a constant slope of 3. Since the slope does not change, there are no critical numbers in this part of the function. However, understanding critical numbers for the other part, where \( x > 1 \), we set the derivative equal to zero, \( -2x = 0 \). Even though the solution here is \( x = 0 \), it does not fall within the interval where this derivative applies, and hence we have no critical numbers for this piecewise function. Knowing how to find and interpret these critical values is essential for analyzing functions and their graphs.

Increasing and Decreasing Intervals

The increasing and decreasing intervals of a function tell us where the output value of the function is getting larger or smaller as the input value increases. To determine these intervals, we look at the sign of the first derivative of the function. If the first derivative is positive on an interval, the function is increasing; if it is negative, the function is decreasing.

In the exercise given, the derivative for \( x \leq 1 \) is a constant 3, which is always positive. Thus the function is increasing on the interval \( (-\infty, 1] \). For \( x > 1 \), the derivative is \( -2x \), which is always negative, leading to the function decreasing on the interval \( (1, \infty) \). It is critical for students to feel confident in determining these intervals, as it helps with graphing the function and understanding its overall behavior.

Piecewise Functions

A piecewise function is defined by different expressions depending on the interval of the input value. It’s like having multiple functions glued together at certain points to form a new, composite function. They can be tricky because each piece may behave differently and has to be considered separately.

In our example, the function \(f(x)\) is defined by one linear equation for \( x \leq 1 \) and a different, quadratic equation for \( x > 1 \). To understand a piecewise function, you must evaluate each piece independently over its specific domain before looking at the function as a whole. The point where the function changes from one piece to another, such as \( x = 1 \) in this exercise, can often be of special interest when analyzing the function’s graph. Piecewise functions can model complex scenarios and are widely used in various fields, such as physics and economics.