Honors Pre-Calculus 3.6 Notes: Critical Points and Extrema Day 1

Honors Pre-Calculus 3.6 Notes: Critical Points and Extrema – day 1

ACT Practice

A, B, C, and D are points on a line, with D the midpoint of . The lengths of are 10, 2, and 12, respectively. What is the length of ?

a) 2 b) 4 c) 6 d) 10 e) 12

Learning Targets:

·  I can find critical points of a function.

·  I can classify the critical points of a function.

Critical Points: points on a graph at which a line drawn tangent to the curve is horizontal or vertical. Critical points will be one of the following: Maximum, Minimum, or Point of Inflection (a kink or change of curvature).

Extrema: general term for maxima and minima

The maximum or minimum VALUE is always “y”.

The LOCATION of the maximum or minimum is always the “x”.

Ex 1. The function f(x)= 5x4 - 10x2 - 20x + 7 has extrema at 1.3. Name and classify the extrema of the function.

Check both sides of the extrema to determine whether it is a min or max.

f(1.2) = -21.032 f(1.3) = -21.6195 f(1.4) = -21.392

Notice the function decreases in value and then increases. This is an absolute min (1.3, -21.6)

Ex 2. Locate the extrema for f(x) = . Name and classify the extrema of the function.

relative min (-2,0) (2,0)

Relative max (0,4)

You Try: Locate the extrema for f(x) = x3 - 8x + 3. Name and classify the extrema of the function.

relative min (1.6,-5.7) Relative max (-1.6,11.7)

Ex 3. The function f(x) = 3x4 – 4x3 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; 3(0)4 – 4(0)3 = 0 inflection

x = 1; 3(1)4 – 4(1)3 = -1 minimum

You Try: The function f(x) = 2x5 - 5x4 - 10x3 has critical points at x = -1, x = 0, and x = 3. Determine whether each of these critical points is the location of a maximum, a minimum, or a point of inflection.

x = -1; 2(-1)5 – 5(-1)4 – 10(-1)3=3 maximum

x = 0; 2(0)5 – 5(0)4 – 10(0)3 = 0 inflection

x = 3; 2(3)5 – 5(3)4 – 10(3)3 = -189 minimum

Honors Pre-Calculus 3.6 Notes: Critical Points and Extrema – day 2

ACT Practice

A right triangle has its vertices at the points (0,0), (5,0), and (0,3). What is the area of this triangle?

a) 4.0 b) 7.5 c) 8.0 d) 8.5 e) 15.0

Optimization Problems

Ex 4. A small business owner employing 15 people hires an analyst to help the business maximize profits. The analyst gathers data and develops the mathematical model P(x) = . In this model, P is the owner’s monthly profits, in dollars, and x is the number of employees. The model has critical points at x = 22 and x = 46.

a) Determine which, if any, of these critical points is a maximum.

x = 22; =9357.333 max

x = 46; = 7053.333

b) What does the critical point suggest to the owner about business operations?

She makes more money with fewer employees

You Try: One hour after x milligrams of a particular drug are given to a person, the rise in body temperature T(x), in degrees Fahrenheit, is given by T(x) = . The model has critical points at x = 4.5.

a) Determine which, if any, of these critical points is a maximum.

x = 4.5; = 2.25

b)  Why should a doctor be aware of this critical point?

So he doesn’t make the person overdose on medication