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 & Extrema Objective:

Critical Points  Critical Points – points on a graph in which a line drawn tangent to the curve is horizontal or vertical  Maximum  Minimum  Point of Inflection

Maximum/Minimum  Tangent lines have a slope=0

Relative Extrema  A 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

Testing for Critical Points let x = a be the critical point for f(x) h is a small value greater than zero Maximum f(a – h) < f(a) f(a + h) < f(a) Minimum f(a – h) > f(a) f(a + h) > f(a) (a, f(a)) (a+h, f(a+h)) (a-h, f(a-h)) h (a, f(a)) h (a-h, f(a-h))(a+h, f(a+h))

Testing for Critical Points let x = a be the critical point for f(x) h is a small value greater than zero Point of Inflection f(a – h) > f(a) f(a + h) < f(a) Point of Inflection f(a – h) < f(a) f(a + h) > f(a) (a, f(a)) (a-h, f(a-h)) (a+h, f(a+h)) h hh 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. x=0 is a point of inflection; x=1 is a minimum

Sources Math First - Massey University. Massey University, n.d. Web. 21 Sept Mrs. Phelps' Math Page. N.p., n.d. Web. 21 Sept Calculus II. Scientificsentence., Web. 21 Sept