"every x over ℝ" - this shows up as a box in my browser.truongson243 said:Homework Statement
For every x over ℝ, find min, max of following expression
I'm assuming that you wrote the right side correctly, where there are three terms, with 1 being a term by itself. Rewrite the right side as a single rational expression. What is the least common denominator?truongson243 said:Homework Equations
y=8x4/(x2+1)2+4x/(x2+1)+1
truongson243 said:The Attempt at a Solution
I've graphed it but it's really complex. How can I find that points? Thank you so much :)
truongson243 said:Homework Statement
For every x over ℝ, find min, max of following expression
Homework Equations
y=8x4/(x2+1)2+4x/(x2+1)+1
The Attempt at a Solution
I've graphed it but it's really complex. How can I find that points? Thank you so much :)
Finding the minimum and maximum values of a function allows us to understand the behavior and characteristics of the function, such as its range, critical points, and overall shape. This information is important in many areas of science, including data analysis, optimization, and modeling.
The most common method is to take the derivative of the function and set it equal to zero to find the critical points. Then, we use the second derivative test or other techniques to determine if the critical points are minimum or maximum values. In some cases, we may also use graphing or numerical methods to approximate the minimum and maximum values.
No, it depends on the complexity and type of the function. Some functions may not have a well-defined minimum or maximum value, while others may have infinitely many minimum or maximum values. In some cases, we may not be able to find the exact values but can approximate them.
The minimum value of a function corresponds to the lowest point on its graph, while the maximum value corresponds to the highest point. If the function is continuous and defined on a closed interval, the minimum and maximum values will occur at either the endpoints of the interval or at critical points within the interval.
In general, no. A function can have multiple critical points that correspond to the same minimum or maximum value. However, if the function is strictly increasing or decreasing, then the minimum and maximum values will be unique. Additionally, if the function is defined on a closed interval and has a continuous graph, then the minimum and maximum values will be unique.