Monotonic Function: Derivative and Interval Analysis for f(x) = x^2 + x + 1

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In summary, We can calculate the derivative of the given function y(x) = x^2 + x + 1 to be f'(x) = 2x + 1. When x = -1/2, the derivative is equal to 0. The function is decreasing on the interval (-∞, -1/2] and increasing on the interval [-1/2, ∞). At x = -1/2, there is an absolute minimum of -3/4. Overall, the function is monotonic increasing on the interval [-1/2, ∞) and monotonic decreasing on the interval (-∞, -1/2].
  • #1
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Homework Statement


function [tex]y(x) = x^2 + x + 1[/tex]

The Attempt at a Solution


I count derivative: [tex]f^{\prime} (x) = 2x + 1[/tex] and now [tex]f^{\prime (x) = 0[/tex] when [tex]x=-\frac{1}{2}[/tex] and how to describe monotonic now? [tex]f(x)[/tex] is decreasing for [tex]x \in \left(- \infty; -\frac{1}{2}\right][/tex] or [tex]x \in \left(- \infty; -\frac{1}{2}\right)[/tex]? open or closed interval? and now increasing for what [tex]x[/tex]?
 
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  • #2
i'd say two open intervals
f(x) is decreasing for [itex]x \in \left(- \infty, -\frac{1}{2}\right) [/itex]
f(x) is increasing for [itex]x \in \left(-\frac{1}{2}\right, \infty \right) [/itex]

and neither at [itex]x = -\frac{1}{2} [/itex]
 
  • #3
I disagree.

f(x) has an absolute minimum of ‒3/4 at x = ‒1/2.

f(x)>f(‒1/2) for x > ‒1/2, so f(x) is monotonic increasing on [‒1/2, +∞) .

Similarly, f(x) is monotonic decreasing on (‒∞ , ‒1/2] .
 

1. What is a monotonic function?

A monotonic function is a mathematical function that either consistently increases or decreases in value as its input variable increases. In other words, the function's derivative is either always positive or always negative, indicating a constant rate of change.

2. What is the derivative of f(x) = x^2 + x + 1?

The derivative of f(x) = x^2 + x + 1 is f'(x) = 2x + 1. This can be found by using the power rule and the sum rule of differentiation, which state that the derivative of x^n is nx^(n-1) and the derivative of a sum of functions is the sum of their individual derivatives.

3. How do you determine the intervals where f(x) = x^2 + x + 1 is increasing or decreasing?

To determine the intervals of increase or decrease for f(x) = x^2 + x + 1, we can use the first derivative test. This involves finding the critical points of the function (where f'(x) = 0 or is undefined) and plugging them into the first derivative. If the first derivative is positive at a critical point, the function is increasing in that interval. If the first derivative is negative, the function is decreasing.

4. Is f(x) = x^2 + x + 1 a strictly monotonic function?

No, f(x) = x^2 + x + 1 is not a strictly monotonic function because it has both increasing and decreasing intervals. It is only a monotonic function in the sense that it consistently increases or decreases over each interval, but it is not strictly increasing or strictly decreasing over its entire domain.

5. How can monotonicity and the derivative be used to analyze the behavior of f(x) = x^2 + x + 1?

By analyzing the monotonicity and derivative of f(x) = x^2 + x + 1, we can determine important characteristics of the function such as its intervals of increase and decrease, its maximum and minimum values, and its concavity (whether it is convex or concave). This information can be used to graph the function and make predictions about its behavior and relationships to other functions.

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