First derivative test

In calculus, the first derivative test is used to find local maxima and minima of a function. The first derivative test can be used to determine intervals of increase and intervals of decrease.

Critical Points (and how to find them)

Critical points are values in the domain of a function where its derivative is 0 or undefined ([math]\displaystyle{ f'(x)=0 }[/math] or [math]\displaystyle{ f'(x) }[/math] does not exist).

To find the critical points of a function, the first derivative of the function is computed and set equal to zero. Next, the zeroes of this equation are solved for. If is is a rational function ([math]\displaystyle{ f(x)=\dfrac{a}{b} }[/math], where [math]\displaystyle{ b \neq 0 }[/math]) the values that will cause the denominator to be equal to 0 are found. This is done because this will make the function undefined. In this case, the equation is not differentiable at that point. These are all of the critical numbers.

These numbers into put into the original function to find the coordinate of these critical points.

Example

If: [math]\displaystyle{ f(x)=\dfrac{1}{3}x^3-\dfrac{5}{2}x^2+6x }[/math]

[math]\displaystyle{ f'(x)=x^2-5x+6 }[/math]

[math]\displaystyle{ (x-3)(x-2)=0 }[/math]

[math]\displaystyle{ x=3 }[/math] and [math]\displaystyle{ x=2 }[/math]

[math]\displaystyle{ f(3)=\dfrac{1}{3}(3)^3-\dfrac{5}{2}(3)^2+6(3)=9-\frac{45}{2}+18=\dfrac{9}{2} }[/math]

[math]\displaystyle{ f(2)=\dfrac{1}{3}(2)^3-\dfrac{5}{2}(2)^2+6(2)=\frac{8}{3}-\frac{20}{2}+12=\dfrac{14}{3} }[/math]

So, the two critical points are [math]\displaystyle{ (3,\dfrac{9}{2}) }[/math] and [math]\displaystyle{ (2,\dfrac{14}{3}) }[/math].

First derivative test

The intervals between points must be considered to know if the points are local maximas, minimas, or neither. To do this, a value from each interval is put into the first derivative of the function. If the value is positive, then it is an interval of increase. If the value is negative, then it is an interval of decrease. If two neighboring intervals change signs, that point is either a local maxima or minima. If the first interval is positive and the second interval negative, the critical point is a local maxima. If the first interval is negative and the second interval positive, the critical point is a local minima. If both intervals are positive or both intervals are negative, the critical point is not a local maxima or minima.

The first derivative test is used in calculus optimization problems.