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

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

   Hint (a):

   $f''(x)$ tells us the same thing about $f′(x)$ that $f′(x)$ tells us about $f(x)$.

2. Write equations for $f′(x)$ and $f''(x)$ if $f(x)=x^3+3x^2-9x+2$.

   Answer (b):

   $f′(x)$: $3x^2+6x-9$
   $f''(x)$:$6x + 6$

3. Is $f$ getting steeper or less steep at $x = 1$? At $x = -2$? Use your derivatives from part (b) to 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 of $f''(1)$ and $f''(-2)$ can be used to determine concavity at $x = 1$ and $x = -2$. Where is $f$ concave up? Where is $f$ 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.