Question

Sketch a graph of a function \(f\) having the given characteristics. (There are many correct answers.) $$ \begin{aligned} &f(-2)=f(0)=0 \\ &f^{\prime}(x)>0 \text { if } x<-1 \\ &f^{\prime}(x)<0 \text { if }-10 \text { if } x>0 \\ &f^{\prime}(-1)=f^{\prime}(0)=0 \end{aligned} $$

Short answer

Without an explicit function, we can't provide an exact graphing. But one can sketch a graph that matches the given conditions. The graph should pass through the points \((-2,0)\) and \((0,0)\), and increase for \(x<-1\) and \(x>0\), and decrease for \(-1<x<0\). It should also have horizontal tangents at \(x=-1\) and \(x=0\).

Step-by-step solution

Plot the given Points

Begin by plotting the points which the function attains the value 0, which are \((-2,0)\) and \((0,0)\). We also note that at these points, the function changes from increasing to decreasing or from decreasing to increasing due to changes in the sign of the derivative.

Sketch the regions

The derivative of function \(f\) is positive for \(x<-1\) and \(x>0\), which means the function is increasing on the intervals \((-\infty, -1)\) and \((0, +\infty)\). The derivative is negative for \(-1<x<0\), meaning the function is decreasing on the interval \((-1,0)\). Sketch these regions according to these properties.

Complete the sketch

Combine the plots from the previous two steps to produce a complete sketch of the function \(f\). The graph should be a curve passing through the points \((-2,0)\) and \((0,0)\) and should adhere to the properties of increase and decrease described.

Key concepts

Function Characteristics

Understanding the characteristics of a function is essential when attempting to sketch its graph. These characteristics provide crucial information about the behavior of the function across different intervals. In this exercise, we start by identifying two critical points, namely \((-2, 0)\) and \((0, 0)\). These points tell us where the function intersects the x-axis, which are often referred to as "zeros" of the function.
Another important characteristic provided is how the function behaves around these zeros, specifically its increasing and decreasing nature in particular intervals. This information, coupled with the derivative analysis, helps us predict the shape of the graph and how the function behaves between and beyond these points.

Derivative Sign

The sign of the derivative \(f'(x)\) gives us powerful insights into the function's behavior. When the derivative is positive \((f'(x) > 0)\), the function is increasing. Conversely, when the derivative is negative \((f'(x) < 0)\), the function is decreasing.
Given this problem, the derivative \(f'(x)\) is positive when \(x < -1\) and \(x > 0\). This means that the function is rising in these intervals. Meanwhile, \(f'(x)\) is negative between \(-1\) and \(0\), indicating that the function descends between these points. Such information helps us visualize the upward and downward trends in a sketched graph.

Critical Points

Critical points occur where the derivative \(f'(x)\) is zero or undefined. In this exercise, these points are given at \(x = -1\) and \(x = 0\).
These are crucial since they often indicate local maxima, minima, or points of inflection where the direction of the function changes. At these points, the function can switch from increasing to decreasing, or vice versa. In the graph sketching, recognizing these points allows us to accurately depict the function's turnarounds or flattening points.

Increasing and Decreasing Intervals

Identifying intervals where a function is increasing or decreasing is foundational in graph sketching. For our exercise, we see that:

• The function is increasing where \(f'(x) > 0\), specifically on intervals \((-\infty, -1)\) and \((0, +\infty)\).
• The function is decreasing where \(f'(x) < 0\), which is within the interval \((-1, 0)\).

Completing a sketch requires plotting these regions with the accurately represented behavior. Collectively, this strengthens our understanding and visualization of the function's overall shape and direction across its domain.