Warmup

If y varies directly as the square of x and inversely as z and y =36 when x =12 and z =8, find x when y=4, and z=32

3.6 – Critical Points & ExtremaObjective:

Critical Points

Critical Points – points on a graph in which a line drawn tangent to the curve is horizontal or verticalMaximumMinimumPoint of Inflection

Maximum/Minimum

Tangent lines have a slope=0

Relative ExtremaA maximum/minimum of a function

in a specific interval.It is not necessarily the max/min for

the entire function

Point of Inflection

Not a maximum or minimum“Leveling-off Point”When a tangent line is drawn here, it

is vertical – slope is undefined

Absolute Extrema

Extrema – the general term of a maximum or minimum.

Absolute Extrema – the greatest/smallest value of a function over its whole domain

Examples

Locate the extrema for the graph. Name and classify the extrema of the function.

Use your graphing calculator to graph

then determine and classify its extrema

38)( 3 xxxf

Testing for Critical Pointslet x = a be the critical point for f(x)h is a small value greater than zero

Maximumf(a – h) < f(a)f(a + h) < f(a)

Minimumf(a – h) > f(a)f(a + h) > f(a)

(a, f(a))

(a+h, f(a+h))(a-h, f(a-h))

h h

(a, f(a))

h h

(a-h, f(a-h)) (a+h, f(a+h))

Testing for Critical Pointslet x = a be the critical point for f(x)h is a small value greater than zero

Point of Inflectionf(a – h) > f(a)f(a + h) < f(a)

Point of Inflectionf(a – h) < f(a)f(a + h) > f(a)

(a, f(a)) (a, f(a))(a-h, f(a-h))

(a-h, f(a-h))

(a+h, f(a+h))(a+h, f(a+h))

h

h h

h

Example

The function has critical points at x=0 and x =1. Determine whether each of these critical points is the location of a maximum, a minimum, or a point of inflection.

34 43)( xxxf

x=0 is a point of inflection; x=1 is a minimum

Sources

Math First - Massey University. Massey University, n.d. Web. 21 Sept. 2013. <http://mathsfirst.massey.ac.nz/Calculus/SignsOfDer/POI.htm>.

Mrs. Phelps' Math Page. N.p., n.d. Web. 21 Sept. 2013. <http://phelpscchs.pbworks.com>.

Calculus II. Scientificsentence., 2010. Web. 21 Sept. 2013. <http://scientificsentence.net/ Equations/CalculusII/index.php?key=yes&Integer=theorems_analysis>.