Problem 16 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. $$ f(x)=2 x^{2}+x+1 $$

Step-by-Step Solution

Answer

The function is increasing in the interval \((-\frac{1}{4}, +\infty)\) and decreasing in the interval \((-\infty, -\frac{1}{4})\).

Step 1: Find the derivative of the function

To find the derivative of the function, use the power rule on each term: $$ \frac{d}{dx} (2x^2 + x + 1) $$ Apply the power rule for each term: $$ \frac{d}{dx} (2x^2) + \frac{d}{dx} (x) + \frac{d}{dx} (1) $$ $$ 4x + 1 $$

Step 2: Find the critical points

Set the derivative equal to 0 and solve for x: $$ 4x + 1 = 0 $$ $$ x = -\frac{1}{4} $$ So, the critical point occurs at x = -1/4.

Step 3: Test intervals around the critical point

Now, we'll test the intervals around the critical point to see if the function is increasing or decreasing: Interval 1 (when x < -1/4): Select a point in the interval, such as x = -1. Plug this value into the derivative and determine whether the result is positive or negative: $$ 4(-1) + 1 = -3 $$ Since the result is negative, the function is decreasing in this interval. Interval 2 (when x > -1/4): Select a point in the interval, such as x = 0. Plug this value into the derivative and determine whether the result is positive or negative: $$ 4(0) + 1 = 1 $$ Since the result is positive, the function is increasing in this interval.

Step 4: Write the final answer

Based on our analysis of the intervals, we can conclude that the function is increasing in the interval \((-\frac{1}{4}, +\infty)\) and decreasing in the interval \((-\infty, -\frac{1}{4})\).

Key Concepts

Derivative

The derivative of a function is a fundamental concept in calculus. It measures how a function changes as its input changes. In simpler terms, the derivative tells us the rate at which one quantity changes in relation to another. For the function we are considering, which is \( f(x) = 2x^2 + x + 1 \), you want to find how this function behaves as \( x \) changes.

To find the derivative, you use rules like the power rule, which says that the derivative of \( x^n \) is \( nx^{n-1} \). So, for \( 2x^2 \), the derivative is \( 4x \). For \( x^1 \), the derivative is 1. Constants like \( 1 \) have a derivative of 0. Therefore, the derivative of this function is \( 4x + 1 \).

Derivatives help in identifying critical points, intervals of increase and decrease, and provide a deep understanding of the function's behavior.

Critical Points

Critical points are places on the graph of a function where the function's derivative is zero or undefined. These points are significant because they can correspond to local maxima or minima—think of them like the peaks and valleys of a mountain range.

To find the critical points of \( f(x) = 2x^2 + x + 1 \), we set the derivative \( 4x + 1 \) equal to zero and solve for \( x \):

• \( 4x + 1 = 0 \)
• Solving for \( x \), we get \( x = -\frac{1}{4} \)

This means there is a critical point at \( x = -\frac{1}{4} \). At this point, the behavior of the function can change from increasing to decreasing, or vice versa. Therefore, critical points are crucial in the analysis of function behavior.

Intervals of Increase and Decrease

Intervals of increase and decrease tell us where a function is going uphill or downhill, respectively. To find these intervals, we look at the derivative and test points on either side of the critical points.

For \( f(x) = 2x^2 + x + 1 \), the derivative is \( 4x + 1 \). We've already found the critical point at \( x = -\frac{1}{4} \). We now examine the sign of \( 4x + 1 \) on intervals around this point:

• For \( x < -\frac{1}{4} \), plug in a value such as \( x = -1 \): \( 4(-1) + 1 = -3 \). The negative result indicates that the function is decreasing in this interval.


• For \( x > -\frac{1}{4} \), plug in a value such as \( x = 0 \): \( 4(0) + 1 = 1 \). The positive result tells us the function is increasing in this interval.

Therefore, the function is decreasing on the interval \(( -\infty, -\frac{1}{4} )\) and increasing on the interval \(( -\frac{1}{4}, +\infty )\). Understanding these intervals helps in plotting and interpreting the behavior and shape of the function's graph.