AP Calculus AB / Unit 5: Analytical Applications of Differentiation
5.7 Using the Second Derivative Test to Determine Extrema
Notes
At a critical point $c$ with $f'(c)=0$:
- $f''(c)>0$: local min.
- $f''(c)<0$: local max.
- $f''(c)=0$: the test is inconclusive; use the first derivative test.
Practice
$f'(3)=0$ and $f''(3)=-2$. At $x=3$, $f$ has:
Concave down at a horizontal tangent.
$f(x)=x^3-12x$. At $x=2$, the second derivative test shows:
$f'(2)=0$ and $f''(2)=12>0$.
If $f'(c)=0$ and $f''(c)=0$:
Check $x^3$, $x^4$, $-x^4$: all three cases are possible.
Ready for more? Take the Unit 5 test.