(A) Find all critical values of f, compute their average, and enter it below. Note: If there are no critical values, enter -1000. Average of critical values =_______________. (B) Use interval notation to indicate where f(x) is increasing. If you have extra boxes, fill each in with an 'x'. Increasing: ______________________ (C) Use interval notation to indicate where f(x) is decreasing.
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
Suppose that
(A) Find all critical values of f, compute their average, and enter it below.
Note: If there are no critical values, enter -1000.
Average of critical values =_______________.
(B) Use interval notation to indicate where f(x) is increasing.
If you have extra boxes, fill each in with an 'x'.
Increasing: ______________________
(C) Use interval notation to indicate where f(x) is decreasing.
Decreasing: ____________________
(D) Find the x-coordinates of all
Note: If there are no local maxima, enter -1000.
Average of x values =___________________
(E) Find the x-coordinates of all
Note: If there are no local minima, enter -1000.
Average of x values =____________________
(F) Use interval notation to indicate where f(x) is concave up.
Concave up: ____________________
(G) Use interval notation to indicate where f(x) is concave down.
Concave down: ___________________
(H) Find all inflection points of f, compute their average, and enter it below.
Note: If there are no inflection points, enter -1000.
Average of inflection points =_________________________
(I) Find all horizontal asymptotes of f, compute the average of the y values, and enter it below.
Note: If there are no horizontal asymptotes, enter -1000.
Average of horizontal asymptotes =_________________________
(J) Find all vertical asymptotes of f, compute the average of the x values, and enter it below.
Note: If there are no vertical asymptotes, enter -1000.
Average of vertical asymptotes =________________________
(K) Use all of the preceding information to sketch a graph of f. When you're finished, enter a "1" in the box below.
Graph Complete:___________________.
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