below is the plot of a function f(x). which is the most accurate table for the derivative function f(x)?

Explanation:

Step1: Recall derivative - slope relationship

The derivative $f^{\prime}(x)$ at a point is the slope of the tangent line to the curve $y = f(x)$ at that point.

Step2: Analyze slope at $x=-3$

At $x = - 3$, the function $y = f(x)$ has a horizontal tangent. So, $f^{\prime}(-3)=0$.

Step3: Analyze slope at $x = 1$

The function is increasing at $x = 1$, and the slope of the tangent line seems to be around $0.6 - 0.8$.

Step4: Analyze slope at a local - extremum

At a local - extremum (where the function changes from increasing to decreasing or vice - versa), the slope of the tangent line is $0$. There is a local - extremum between $x = 1$ and $x = 2$, so there is a point in this interval where $f^{\prime}(x)=0$.

Step5: Analyze slope at $x = 2$

The function is increasing at $x = 2$, and the slope of the tangent line is positive. It seems to be around $0.8 - 1$.

Step6: Analyze slope at $x = 3$

The function is increasing at $x = 3$, and the slope of the tangent line is positive and larger than at $x = 2$. It seems to be around $1.8$.

Looking at the tables, the first table has the correct values based on our analysis: $f^{\prime}(-3)=0$, a non - zero positive value for $f^{\prime}(1)$ (around $0.6$), $f^{\prime}(x) = 0$ at a point between $x = 1$ and $x = 2$, a non - zero positive value for $f^{\prime}(2)$ (around $0.8$), and a larger positive value for $f^{\prime}(3)$ (around $1.8$).

Answer:

The first table (the one with $x=-3,f^{\prime}(-3)=0;x = 1,f^{\prime}(1)=0.6;$ a value of $x$ between $1$ and $2$ with $f^{\prime}(x)=0;x = 2,f^{\prime}(2)=0.8;x = 3,f^{\prime}(3)=1.8$)