x 1 4 6 7 9 12\nf(x) -25 -33 -19 -37 -31 -24\nwhat is the best estimate for f(2) we can make based on this…

Explanation:

Step1: Use the symmetric - difference quotient formula

The symmetric - difference quotient for estimating the derivative $f^{\prime}(a)$ is $f^{\prime}(a)\approx\frac{f(a + h)-f(a - h)}{2h}$. Since we want to estimate $f^{\prime}(2)$ and the closest values in the table to $x = 2$ are $x = 1$ and $x = 4$. Here, $a = 2$, $h = 1$, $f(a - h)=f(1)=-25$ and $f(a + h)=f(4)=-33$.

Step2: Calculate the estimate

Substitute the values into the symmetric - difference quotient formula: $f^{\prime}(2)\approx\frac{f(4)-f(1)}{4 - 1}=\frac{-33-(-25)}{3}=\frac{-33 + 25}{3}=\frac{-8}{3}\approx - 2.67$.

Answer:

C. $-2.67$