Question

question 5 (1 point) retake question
the graphs a (blue), b (red), and c (green) are the graphs of a function f(x) and its derivatives, f(x) and f(x). match the graph to the corresponding function.

Explanation:

Step1: Analyze the relationship between \(f(x)\), \(f^{\prime}(x)\), and \(f^{\prime\prime}(x)\)

• The derivative \(f^{\prime}(x)\) represents the slope of the function \(f(x)\).
• The second - derivative \(f^{\prime\prime}(x)\) represents the concavity of \(f(x)\) (or the slope of \(f^{\prime}(x)\)).

Step2: Look for key features

• When \(f^{\prime}(x)=0\), the function \(f(x)\) has a local extremum (maximum or minimum).
• When \(f^{\prime\prime}(x)=0\), the function \(f(x)\) has an inflection point (the concavity changes).

• Assume \(A\) is \(f(x)\). Then look at the slope of \(A\). The slope of \(A\) (which is \(f^{\prime}(x)\)) has zeros (where the slope of \(A\) is zero). If we check the other two graphs \(B\) and \(C\), we can match the zeros of \(f^{\prime}(x)\) (slope of \(A\)) with the \(x -\)intercepts of one of the other graphs.
• Also, look at the concavity of \(A\). The concavity of \(A\) (which is determined by \(f^{\prime\prime}(x)\)) changes. The sign of \(f^{\prime\prime}(x)\) (positive for concave - up, negative for concave - down) should correspond to the sign of one of the remaining graphs.

• Another approach: The degree of oscillation. The second - derivative \(f^{\prime\prime}(x)\) has the “highest frequency” of the three functions (in terms of the number of local extrema). If we assume \(C\) is \(f^{\prime\prime}(x)\) (more local extrema than \(B\)), \(B\) is \(f^{\prime}(x)\) and \(A\) is \(f(x)\)

Answer:

\(A:f(x)\), \(B:f^{\prime}(x)\), \(C:f^{\prime\prime}(x)\)