Question

10 points quarter 1 week 6 this warm - up is due friday oct 10 during the first five minutes of class. answer each problem completely showing all work in the space provided and placing your answer in the space provided. if you do not show any work for a problem then you will not receive any credit for that problem. this assignment will be recorded in the classwork category. if you are absent in friday, this is due the first day you are back in class. this assignment is not eligible for retake. please print your name name / date 1. the function values are positive, and roc is negative. the graph of f(x) is given to the right. identify all intervals where the following occurs: 2. the function values are decreasing, and roc is increasing. 3. the roc is positive, and increasing concave up 4. the roc is decreasing, and the function is decreasing 5. the change in roc is negative, and the roc is positive, and the function is negative.

Explanation:

Step1: Recall function - related concepts

The rate of change (ROC) of a function is related to its derivative. A positive ROC means the function is increasing, and a negative ROC means the function is decreasing. Concavity is related to the second - derivative. An inflection point is where the concavity changes.

Step2: Analyze the graph visually

1. Intervals of positive function values and negative ROC:

• Look for the parts of the graph where the function \(y = f(x)\) is above the \(x\) - axis (positive function values) and the slope of the tangent line to the curve is negative. From the graph, we can see that in the interval \((2,3)\), the function is above the \(x\) - axis (positive function values) and the curve is sloping downwards (negative ROC).

1. Intervals where the function is decreasing and ROC is increasing:

• A function is decreasing when its first - derivative is negative, and the ROC (first - derivative) is increasing when the second - derivative is positive. Looking at the graph, in the interval \((-3,-2)\), the function is sloping downwards (decreasing) and the curve is becoming less steep in the negative - slope direction (ROC is increasing).

1. Inflection points:

• Inflection points are where the concavity changes. Visually, we look for the points where the curve changes from being concave up to concave down or vice - versa. From the graph, the inflection points occur at \(x=-2\) and \(x = 2\).

1. Global minimum:

• The global minimum is the lowest point on the entire graph. From the graph, the global minimum occurs at \(x = 3\) and the function value at this point is \(-3\).

1. Intervals for the given ROC and function sign conditions:

• We need to find an interval where the change in ROC is negative (second - derivative is negative), the ROC is positive (first - derivative is positive), and the function is negative. Looking at the graph, in the interval \((-\infty,-3)\), the curve is sloping upwards (positive ROC), the slope is becoming less steep (negative change in ROC), and the function is below the \(x\) - axis (negative function values).

Answer:

1. Interval where function values are positive and ROC is negative: \((2,3)\)
2. Interval where the function is decreasing and ROC is increasing: \((-3,-2)\)
3. Inflection points: \(x=-2,x = 2\)
4. Global minimum: \(x = 3\), \(y=-3\)
5. Interval where the change in ROC is negative, ROC is positive, and the function is negative: \((-\infty,-3)\)