Use the given function to find each of the following in parts a – d. In part e, use what you have gathered from the previous parts to graph the function. f(x)=-x^3+3x^2+9x-27 a)Find the x-intercepts and y-intercept and write your answers as ordered pairs to be plotted on your graph in part e. b)Find the critical numbers and write your answers as ordered pairs to be plotted on your graph in part e. c)Find the relative extrema (by using either the 1st or 2nd derivative test) and write each as an ordered pair to be plotted on your graph in part e. d)Find the point of inflection and determine the intervals where the function is concave up or concave down. Write your point of inflection as an ordered pair to be plotted on your graph in part e. e)Sketch the graph, and be sure to label all points found in parts a-d. Also, be sure to label your axes and the equation of function. Use the chart if its helpful to keep track of what you had found in the previous parts.
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.
Use the given function to find each of the following in parts a – d. In part e, use what you have gathered from the previous parts to graph the function.
f(x)=-x^3+3x^2+9x-27
a)Find the x-intercepts and y-intercept and write your answers as ordered pairs to be plotted on your graph in part e.
b)Find the critical numbers and write your answers as ordered pairs to be plotted on your graph in part e.
c)Find the relative extrema (by using either the 1st or 2nd derivative test) and write each as an ordered pair to be plotted on your graph in part e.
d)Find the point of inflection and determine the intervals where the function is concave up or concave down. Write your point of inflection as an ordered pair to be plotted on your graph in part e.
e)Sketch the graph, and be sure to label all points found in parts a-d. Also, be sure to label your axes and the equation of function. Use the chart if its helpful to keep track of what you had found in the previous parts.
Trending now
This is a popular solution!
Step by step
Solved in 10 steps with 3 images
