Question

the graph of $f$, the derivative of $f$, is shown below. determine all possible inferences that can be made about $f$ and $f$ when $x = 1$.
note: even though we cannot technically say a function is increasing or decreasing at a point, you should indicate that either function is increasing or decreasing at $x = 1$ if it is doing so in a small interval surrounding $x = 1$.
answer attempt 1 out of 3

Explanation:

Step1: Analyze the value of \(f^{\prime}(1)\)

From the graph of \(y = f^{\prime}(x)\), when \(x = 1\), we can directly observe the \(y\)-value. The \(y\)-value of \(f^{\prime}(x)\) at \(x = 1\) is negative. So \(f^{\prime}(1)<0\).

Step2: Analyze the feature of \(f(x)\) based on \(f^{\prime}(x)\)

According to the first - derivative test, if \(f^{\prime}(x)<0\) on an interval containing \(x = 1\), then the function \(y = f(x)\) is decreasing on that interval. So \(f(x)\) is decreasing at \(x = 1\) (in a small interval around \(x = 1\)).

Step3: Analyze the sign - change of \(f^{\prime}(x)\)

Looking at the graph of \(y = f^{\prime}(x)\) around \(x = 1\), we check if the sign of \(f^{\prime}(x)\) changes. The sign of \(f^{\prime}(x)\) does not change at \(x = 1\). For example, if we consider values slightly less than \(1\) and slightly greater than \(1\), the \(y\)-values of \(f^{\prime}(x)\) (in a small neighborhood) have the same sign (negative).

Answer:

	\(f(1)\)	\(f^{\prime}(1)\)
Feature	Decreasing (in a small interval around \(x = 1\))	Negative (derivative value)
Sign Change	N/A	No sign - change