4.3 - The First and Second Derivative Tests

#calc/concept

The first derivative test

The first derivative test has already been covered in this unit – see 4.1 - Extreme Values of Functions.

• Essentially, the zeroes and undefined points of the first derivative of a function are critical points, and by using a sign chart it’s possible to see whether the point is a min, a max, or neither.
• The reason why this is important is that it provides easy justification for a lot of FRQs.

The second derivative test

• We can see that a function $f(x)$ is concave up if $f^{″}(x)$ is positive and concave down if $f^{″}(x)$ is negative.
  • This means that the points of inflection (where the change in concavity occurs) are at the zeroes of the second derivative.

Testing for local extrema (actual statement of the test)

The second derivative test for local extrema states that:

1. If $f^{\prime }(c)=0$ and $f^{″}(c)<0$, then $f$ has a local maximum at $x=c$.
2. f $f^{\prime }(c)=0$ and $f^{″}(c)>0$, then $f$ has a local minimum at $x=c$.

At points where the first derivative is not defined, you cannot use the second derivative test.