Critical Values

Definition

These are points where:

• $c$ is in the domain of $f$
• $f^{\prime }(c)=0$ or $f^{\prime }(c)$ is undefined

Example with Rational Function

$\frac{(x-2)(x+1)}{(x+4)(x-3)}\ge 0$ The critical values are when the numerator is zero and when the denominator is zero. Critical values: x = 2, x = -1, x = -4, x = 3.

First Derivative Test

If:

• $f(x)$ is has a derivative and is differentiable on $(a,b)$ except possibly at $c$
• If $f^{\prime }(x)$ is positive on $x\in (a,c)$ and negative on $x\in (c,b)$ Then $f$ has a local max on $x=c$
• If $f^{\prime }(x)$ is negative on $x\in (a,c)$ and positive on $x\in (c,b)$ Then $f$ has a local min on $x=c$
• If $f^{\prime }(x)$ is positive for both $x\in (a,c)$ and $x\in (c,b)$ or its negative on $x\in (a,c)$ and $x\in (b,c)$ Then $f$ has no local extremum at x=c