If f(x) = x³ + 1, then give the following. (a) the value of f at x = -1 is (b) the value of f at x = 2 is (c) the net change in the value of f between x = -1 and x = 2 is f )-r(-1) =D
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.
![## Polynomial Functions: Analyzing \( f(x) = x^3 + 1 \)
### Problem Statement
Given the function \( f(x) = x^3 + 1 \), complete the following tasks:
### (a) Calculation at \( x = -1 \)
**Problem:** Find the value of \( f \) at \( x = -1 \).
\[ f( \boxed{-1} ) = \boxed{} \]
### (b) Calculation at \( x = 2 \)
**Problem:** Find the value of \( f \) at \( x = 2 \).
\[ f( \boxed{2} ) = \boxed{} \]
### (c) Net Change in Value
**Problem:** Calculate the net change in the value of \( f \) between \( x = -1 \) and \( x = 2 \).
\[ f( \boxed{2} ) - f(-1) = \boxed{} \]
### Further Explanation
To complete these tasks, perform the following steps:
1. Calculate \( f(-1) \):
\[
f(-1) = (-1)^3 + 1 = -1 + 1 = 0
\]
2. Calculate \( f(2) \):
\[
f(2) = 2^3 + 1 = 8 + 1 = 9
\]
3. Determine the net change:
\[
\text{Net Change} = f(2) - f(-1) = 9 - 0 = 9
\]
The boxed areas should be filled with the respective calculated values:
- \( f(-1) = 0 \)
- \( f(2) = 9 \)
- Net Change \( = 9 \)
### Final Answers
(a) \( f(-1) = 0 \)
(b) \( f(2) = 9 \)
(c) Net Change \( = 9 \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6f4719ec-89c6-49f2-a8fd-c905fe776b01%2F6a4b278b-925c-4531-8668-aa744a7bdb13%2F68vm4cm_processed.jpeg&w=3840&q=75)
Step by step
Solved in 2 steps with 2 images
