Let $f^\prime(x) =\frac { \operatorname { ln } | x - 2 | } { 3 }$ and $f(−3) =\frac { 7 } { 3 }$.

1. Estimate f(−3.1).

   Hint (a):

   Do NOT find an $f(x)$ function. That would be tedious and unnecessary work.

   Steps (a):

   This is another way of wording a very familiar problem:
   'Given $f^\prime(x)$ and a point on $f(x)$, find the tangent line to $f(x)$ at $x = −3$ and use it to estimate the value of $f(−3.1)$.'

2. Find $f^{\prime\prime}(x)$.

   Hint (b):

   Split the absolute value into a piecewise function and differentiate.
   Or, you could recall a special note about the antiderivative of $x^{−1}$: While you can generally leave it as $\operatorname{ln}(x)$, it actually equals $\operatorname{ln}|x|$.

3. Use concavity to determine if your answer to part (a) is an under or over estimate of the actual value of $f(x)$. Justify your answer.

   Hint (c):

   If a function is concave down, all tangent lines will be above the curve.
   If a function is concave up, all tangent lines will be below the curve. Sketch this to verify.