Problem 19 - 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 extrema, and (d) use a graphing utility to confirm your results. $$ f(x)=-2 x^{2}+4 x+3 $$

Step-by-Step Solution

Answer

The critical number of \(f(x) = -2x^2 + 4x + 3\) is \(x = 1\). The function is increasing on the interval \((-∞, 1)\) and decreasing on the interval \((1, ∞)\). There is a relative maximum at \(x = 1\). A graph of the function confirms these results.

Step 1: Finding the derivative

Start by taking the derivative of the function \(f(x)\). The derivative of \(f(x) = -2x^2 + 4x + 3\) is \(f'(x) = -4x + 4\).

Step 2: Finding critical numbers

Next, you need to find the critical numbers of the function. These are the \(x\)-values where the derivative equals to 0 or is undefined. The derivative \(f'(x) = -4x + 4\) is undefined nowhere as it's a polynomial, and its root can be find as follows: \(-4x + 4 = 0\), which simplifies to \(x = 1\). Therefore, the critical number of \(f(x)\) is \(x = 1\).

Step 3: Identifying intervals of increase and decrease

The function \(f(x)\) increases when \(f'(x) > 0\) and decreases when \(f'(x) < 0\). To find where \(f'(x)\) is negative or positive, use a sign chart. The critical number splits the number line into two intervals. Test each interval by choosing a test point in it and evaluate the sign of the derivative at the test point. For \(x < 1\), use \(x = 0\); and for \(x > 1\), use \(x = 2\). The derivative \(f'(0) = 4 > 0\), so \(f(x)\) is increasing on the interval \((-∞, 1)\). The derivative \(f'(2) = -4 < 0\), so \(f(x)\) is decreasing on the interval \((1, ∞)\).

Step 4: Applying the First Derivative Test

The First Derivative Test can be used to identify relative extrema. If \(f'(x)\) changes sign from plus to minus at a critical number, then \(f(x)\) has a relative maximum there. If \(f'(x)\) changes from minus to plus, then \(f(x)\) has a relative minimum there. Since the derivative \(f'(x)\) changes from positive to negative at \(x = 1\), the function \(f(x)\) has a relative maximum at \(x = 1\).

Step 5: Confirming the results graphically

You can verify these results by graphing the function \(f(x) = -2x^2 + 4x + 3\). The graph should show a relative maximum at \(x = 1\), indicating that \(f(x)\) increases on the interval \((-∞, 1)\) and decreases on the interval \((1, ∞)\).

Key Concepts

Critical Numbers

Understanding critical numbers is crucial for analyzing functions. A critical number is a point on the domain of a function where its derivative is either zero or undefined. In simple terms, they're where the function's graph has a horizontal tangent or a sharp corner.

In the given exercise, a critical number is identified by setting the derivative of the function, which is \(f'(x) = -4x + 4\), equal to zero, leading to the critical number at \(x = 1\). It's essential because it represents a potential location for a peak or valley in the graph of the function, in other words, a potential extremum. When given a function like \(f(x) = -2x^2 + 4x + 3\), finding the critical numbers is the first step to mapping its behavior.

Increasing and Decreasing Intervals

Determining when a function is increasing or decreasing is critical for understanding its overall shape and where it achieves its highest and lowest values.

The function is increasing when its slope (the derivative) is positive and decreasing when the slope is negative. In our exercise, by analyzing where the derivative \(f'(x) = -4x + 4\) is positive or negative, we establish that the function is increasing on the interval \( (-\infty, 1) \) and decreasing on \( (1, \infty) \). Understanding these intervals helps us to predict the function's behavior between and beyond critical points and is an essential tool for sketching graphs of functions without a graphing utility.

Relative Extrema

The concept of relative extrema involves finding the points where a function reaches its local maximums and minimums. These are the highest or lowest points in a neighborhood around a particular x-value.

Using the First Derivative Test, if the derivative of a function changes from positive to negative at a critical number, we have a relative maximum. Conversely, if it changes from negative to positive, we have a relative minimum. In our example with the function \(f(x)\), the derivative \(f'(x)\) changes from positive to negative at \(x = 1\), signaling a relative maximum at this point. Recognizing these relative extrema is valuable as they are often significant in the contexts of optimization and modeling situations.

Graphing Utility

A graphing utility is a powerful technological tool that helps in visualizing the shape of functions and confirming analytical findings. After performing steps like finding critical numbers and determining intervals of increase and decrease, a graphing utility can be used to ensure that the visual representation of the function aligns with our calculations.

When graphing \(f(x) = -2x^2 + 4x + 3\), the graphing utility should exhibit the identified patterns — a relative maximum at \(x = 1\), increasing behavior to the left of \(x = 1\), and decreasing behavior to the right of it. Employing graphing utilities reinforces understanding and provides a method to check work, thus enhancing learning and comprehension of calculus concepts.