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Prove that the function$$ f(x) = x^101 + x^51 + x + 1 $$has neither a local maximum nor a local minimum.
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00:59
Wen Zheng
Calculus 1 / AB
Calculus 2 / BC
Chapter 4
Applications of Differentiation
Section 1
Maximum and Minimum Values
Derivatives
Differentiation
Volume
Campbell University
University of Michigan - Ann Arbor
University of Nottingham
Idaho State University
Lectures
04:35
In mathematics, the volume of a solid object is the amount of three-dimensional space enclosed by the boundaries of the object. The volume of a solid of revolution (such as a sphere or cylinder) is calculated by multiplying the area of the base by the height of the solid.
06:14
A review is a form of evaluation, analysis, and judgment of a body of work, such as a book, movie, album, play, software application, video game, or scientific research. Reviews may be used to assess the value of a resource, or to provide a summary of the content of the resource, or to judge the importance of the resource.
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Let's first take the derivative. As we can see, we have positive coefficients and we have powers that are even 150 or both even. Therefore, we know it's always increasing something that's always increasing. Let's say it looks like this or looks like this. It doesn't have a minimum of Maxim because it's always constantly increasing.
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