Calculus / Unit 6: Applications of Derivatives II
6.3 Concavity and the Second Derivative
Notes
The second derivative $f''$ describes how the slope changes.
- $f'' > 0$: concave up (the graph holds water, slopes increasing).
- $f'' < 0$: concave down.
- An inflection point is where concavity changes, so $f''$ changes sign.
- Second derivative test: if $f'(c) = 0$, then $f''(c) > 0$ gives a local min and $f''(c) < 0$ gives a local max.
- Example: $f(x) = x^3 - 6x^2$ has $f'' = 6x - 12$, concave up for $x > 2$, with an inflection point at $x = 2$.
Practice
If $f''(x) > 0$ on an interval, the graph of $f$ is:
A positive second derivative means concave up. Increasing or decreasing depends on $f'$.
Where does $f(x) = x^3 - 6x^2 + 9x$ have an inflection point?
$f'' = 6x - 12$ is 0 at $x = 2$ and changes sign there.
Use the second derivative test on $f(x) = -x^2 + 6x$. What is at $x = 3$?
$f' = -2x + 6 = 0$ at 3, and $f'' = -2$.
Ready for more? Take the Unit 6 test.