You know that the first derivative, $f^\prime(x)$, tells us the slope and the rate of change of $f(x)$.

1. What does the second derivative, $f^{\prime\prime}(x)$, tell you about $f^\prime(x)$? What does $f^{\prime\prime}(x)$ tell you about $f(x)$?

   Hint (a):

   $f^{\prime\prime}(x)$ tells us the same thing about $f^\prime(x)$ that $f^\prime(x)$ tells us about $f(x)$.

2. Find $f^\prime(x)$ and $f^{\prime\prime}(x)$ for $f(x) = x^3 + 3x^2-9x + 2$.

   Answer (b):

   $f^\prime(x)$: $3x^² + 6x-9$
   $f^{\prime\prime}(x)$: $6x + 6$

3. Use these derivatives to find $f^{\prime\prime}( 1 )$ and $f^{\prime\prime}(-2)$. Is $f(x)$ getting steeper or less steep at $x = 1$? At $x =-2$? Explain your reasoning.

   Hint (c):

   Consider this: Even though a slope of $−2$ is steeper than a slope of $−1$, if the slope changes from $−1$ to $−2$, then the slope is decreasing.

4. The values in part (c) can be used to determine concavity. Where is $f(x)$ concave up? Where is $f(x)$ concave down?

   Hint (d):

   Positive values of the 2nd-derivative indicate that the function is concave up while negative values indicate that the function is concave down.