Question

you are given the following graph of the function $f(x)$: find the point where the second derivative changes sign from negative to positive? $x=-2$

Explanation:

Step1: Recall the concept of concavity and second - derivative

The second derivative \(f''(x)\) determines the concavity of the function \(y = f(x)\). If \(f''(x)>0\), the function is concave up; if \(f''(x)<0\), the function is concave down. A point where the concavity changes (from concave down to concave up) is a point where \(f''(x)\) changes sign from negative to positive.

Step2: Analyze the graph for concavity change

Looking at the graph of \(y = f(x)\), we observe the regions of concavity. The function changes from concave down (where the slope of the tangent line is decreasing) to concave up (where the slope of the tangent line is increasing).

Step3: Locate the \(x\) - value of the concavity change

By visual inspection of the graph, we can see that the concavity changes from concave down to concave up at \(x = 1\).

Answer:

\(x = 1\)