Problem 53 - Single Variable Calculus

Question

Sketch the graph of a function having the given properties. $$ \begin{array}{l} f(-1)=0, f^{\prime}(-1)=0 \\ f(0)=1, f^{\prime}(0)=0 \\ f^{\prime}(x)<0 \text { on }(-\infty,-1) \\ f^{\prime}(x)>0 \text { on }(-1, \infty) \\ f^{\prime \prime}(x)>0 \text { on }\left(-\infty,-\frac{2}{3}\right) \cup(0, \infty) \\ f^{\prime \prime}(x)<0 \text { on }\left(-\frac{2}{3}, 0\right) \end{array} $$

Step-by-Step Solution

Answer

The function has a local minimum at \((-1,0)\), a local maximum at \((0,1)\), and an inflection point between \(-2/3\) and \(0\). The graph is continuous, decreasing on \((-\infty, -1)\), increasing on \((-1, \infty)\), concave up on \(\left(-\infty, -\frac{2}{3}\right) \cup (0, \infty)\), and concave down on \(\left(-\frac{2}{3}, 0\right)\). The sketch should depict these characteristics, with the curve passing through the points \((-1,0)\) and \((0,1)\), changing concavity at the inflection point between \(-2/3\) and \(0\).

Step 1: Understand the properties of the function

We are given several pieces of information about the function and its derivatives: 1. The function is continuous since there are no contradictory statements regarding the endpoints or describing discontinuities. 2. We have critical points at \(x=-1\) and \(x=0\) because \(f'(-1)=0\) and \(f'(0)=0\). These are the points where the function's slopes are equal to zero. 3. The function is decreasing between \(-\infty\) and \(-1\) because \(f'(x)<0\) and increasing between \(-1\) and \(\infty\) since \(f'(x)>0\). 4. The function is concave up when \(x<-2/3\) and \(x>0\) because \(f''(x)>0\) and concave down when \(-2/3

Step 1: Mark the critical points on the graph

First, mark the given critical points on the graph: 1. \(f(-1)=0\), which means the graph passes through the point \((-1,0)\). 2. \(f(0)=1\), which means the graph passes through the point \((0,1)\). These two points will be helpful when drawing the shape of the graph.

Step 2: Determine local extrema

Since we know that the function is decreasing on the interval \(-\infty

Step 3: Mark inflection points and the shape of the function

Considering the concavity information of the function given by the second derivative, we can determine the shape of the graph. The function f is concave up when \(x<-2/3\) and \(x>0\), which means there is an inflection point somewhere between \(-2/3\) and \(0\). We don't know the exact point, so we just need to keep that in mind while drawing this part of the graph. As we've identified the concavity, now we can sketch the graph! 1. Draw the curve passing through \((-1,0)\), the local minimum, and concave up on the left side. 2. Draw the curve passing through \((0,1)\), the local maximum, and concave down between \(-2/3\) and \(0\). 3. Continue to draw the curve concave up after \(x=0\). Be sure to draw it smoothly, following the shape indicated by the concavity information. Now we've sketched the graph of the function f(x) following the given properties. Keep in mind that this is a sketch, and there can be multiple valid interpretations depending on the choice of the inflection point and the smoothness of the curve.

Key Concepts

Critical Points

Critical points are crucial in understanding the behavior of a function. They are the points on the graph where the derivative of the function is zero or undefined. This usually corresponds to local peaks, valleys, or points of horizontal tangency.

• In our given problem, the critical points are at \(x = -1\) and \(x = 0\) because \(f'(-1) = 0\) and \(f'(0) = 0\).
• These points might indicate where the function changes direction; however, it's not always guaranteed. We need more information about the function's behavior around these points.

The given problem states critical points by solving the derivative \(f'(x) = 0\), pointing us to where the function has potential extrema (local maxima or minima).

Concavity

The concept of concavity helps us understand how the function bends around points in its domain. Mathematically, it is determined by the second derivative \(f''(x)\).

• If \(f''(x) > 0\), the function is said to be concave up at that point, and the graph resembles a 'U' shape.
• If \(f''(x) < 0\), the function is concave down, resembling an 'n' shape.

For the problem at hand, the graph is:

• Concave up for \(x < -2/3\) and \(x > 0\).
• Concave down for \(-2/3 < x < 0\).

These concavity intervals will help us not only in sketching the graph but also in identifying where the graph might have inflection points. Understanding concavity will enhance our sketch by providing a more precise structure to the curves.

Local Extrema

Local extrema are the local highest and lowest points on a graph within a certain region, known as local maxima and minima, respectively. These are easily found using the critical points in conjunction with the first derivative test.

• In our function, there is a local minimum at \((-1,0)\). Since the function is decreasing before and increasing after this point, \((-1,0)\) turns out to be a local valley.
• Similarly, we have a local maximum at \((0,1)\), appearing as a peak because the function increases before this point and decreases afterward.

Identifying local extrema simplifies the task of sketching complex functions, giving distinct points on the graph where the direction changes significantly.

Inflection Points

Inflection points are points on the graph where the concavity changes from up to down or vice versa. They are where the second derivative does not maintain its sign, often crossing zero.
For the function in this exercise:

• The inflection point should lie within the interval \(-2/3 < x < 0\), as this is where the change in concavity from concave down to concave up happens.
• While the exact inflection point isn't determined here, knowing the interval helps in shaping the curve accurately.

These inflection points are critical to sketching because they show a change in how the graph 'bends,' offering a clear guide on the graph's style. Inflection points don't necessarily correspond to any special heights but rather to changes in the bending manner of the graph.

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Key Concepts

Critical Points

Critical points are crucial in understanding the behavior of a function. They are the points on the graph where the derivative of the function is zero or undefined. This usually corresponds to local peaks, valleys, or points of horizontal tangency.

• In our given problem, the critical points are at \(x = -1\) and \(x = 0\) because \(f'(-1) = 0\) and \(f'(0) = 0\).
• These points might indicate where the function changes direction; however, it's not always guaranteed. We need more information about the function's behavior around these points.

The given problem states critical points by solving the derivative \(f'(x) = 0\), pointing us to where the function has potential extrema (local maxima or minima).

Concavity

The concept of concavity helps us understand how the function bends around points in its domain. Mathematically, it is determined by the second derivative \(f''(x)\).

• If \(f''(x) > 0\), the function is said to be concave up at that point, and the graph resembles a 'U' shape.
• If \(f''(x) < 0\), the function is concave down, resembling an 'n' shape.

For the problem at hand, the graph is:

• Concave up for \(x < -2/3\) and \(x > 0\).
• Concave down for \(-2/3 < x < 0\).

These concavity intervals will help us not only in sketching the graph but also in identifying where the graph might have inflection points. Understanding concavity will enhance our sketch by providing a more precise structure to the curves.

Local Extrema

Local extrema are the local highest and lowest points on a graph within a certain region, known as local maxima and minima, respectively. These are easily found using the critical points in conjunction with the first derivative test.

• In our function, there is a local minimum at \((-1,0)\). Since the function is decreasing before and increasing after this point, \((-1,0)\) turns out to be a local valley.
• Similarly, we have a local maximum at \((0,1)\), appearing as a peak because the function increases before this point and decreases afterward.

Identifying local extrema simplifies the task of sketching complex functions, giving distinct points on the graph where the direction changes significantly.

Inflection Points

Inflection points are points on the graph where the concavity changes from up to down or vice versa. They are where the second derivative does not maintain its sign, often crossing zero.
For the function in this exercise:

• The inflection point should lie within the interval \(-2/3 < x < 0\), as this is where the change in concavity from concave down to concave up happens.
• While the exact inflection point isn't determined here, knowing the interval helps in shaping the curve accurately.

These inflection points are critical to sketching because they show a change in how the graph 'bends,' offering a clear guide on the graph's style. Inflection points don't necessarily correspond to any special heights but rather to changes in the bending manner of the graph.