Prove that the function f(x)=x^101+x^51+x+1 has neither a local maximum nor a local minimum.

Name: Problem 75, Chapter 4: Calculus Early Transcendentals
Uploaded: 2020-06-08T22:43:24-08:00
Duration: 5 min 56 s
Channel: Shakiyla Huggins
Description: Prove that the function f(x)=x^101+x^51+x+1 has neither a local maximum nor a local minimum.

Prove that the function f(x)=x^101+x^51+x+1 has neither a...

Key Concepts

Critical Points in Calculus

Critical points of a function are those points at which the derivative is zero or undefined, and they are typically investigated when searching for local maxima and minima. However, if a function has no critical points, or if its derivative never changes sign, it indicates that there are no local extrema. In the case of strictly monotonic functions, the absence of zero or sign-changing derivative confirms there are no points where a maximum or minimum could occur.

Monotonicity

A function is monotonic if it consistently increases or decreases over its domain. When a function is strictly monotonic, such as strictly increasing, it means that for any two distinct points in its domain, the function values preserve a consistent order. This property ensures that the function cannot have any local maximum or minimum because there is no turning point where the function reverses its trend.

First Derivative Test for Monotonicity

The first derivative of a function provides information about the function's rate of change. If the derivative is strictly positive for all values in the domain, the function is strictly increasing, and if it is strictly negative, the function is strictly decreasing. This test directly relates the sign of the derivative to the absence of local extrema, since a constant sign means the function does not change direction to form a local peak or valley.

Transcript

00:01 Okay guys, so for this question, we have to prove that the function f of x is equal to x to the 101 plus x to the 51 plus x plus one has neither a local maximum nor a local minimum.

00:15 Now, remember, local maximum and local minimum both deal with the first derivative.

00:24 So we need to take the first derivative of our function.

00:26 So let's go ahead and do that now.

00:28 Remember f prime of x is what is used to denote derivative so f prime of x is going to be equal to 101 x to the 100 plus 51 x to the 50 plus 1 now if you don't quite remember how to do that derivative remember you always take your exponent and put it out front and then subtract 1 so that's how the 1001 comes out front and then 100 and 1 minus 1 is 100.

01:00 So that's how we went through our derivative.

01:03 Now, remember that the derivative of x is always just going to be 1, and the derivative of any constant is always going to be just 0.

01:11 So that's why we don't have a derivative out here for our constant of 1.

01:16 All right.

01:17 So now that we have to, excuse me, now that we have the derivative of our function, we need to think about some rules that we know about derivative.

01:26 So i'm going to go ahead and write these rules over here.

01:28 So that you can see them.

01:31 All right, so here are our rules.

01:32 Number one, if your derivative, if f prime of x is greater than zero over a certain interval, then f of x is going to be increasing over that same interval.

01:45 Now, if your derivative or if your f prime of x is less than zero, then f of x is going to be decreasing over that interval.

01:53 And if f prime of x is equal to zero, then f of x is going to be constant.

01:58 So what we need to figure out is, is our f prime of x, is it greater than zero? is it less than zero? is it going to be equal to zero? now, the easiest way to do that is to find a table or to create the table and draw a makeshift graph.

02:16 Okay? so i'm going to use very simple numbers.

02:19 I'm going to go with negative 1, 0, and 1.

02:24 All right, so i'm just going to take each of these numbers and plug them into my function here.

02:29 So if i start with zero, then i'm going to do 101 times zero to the power of 100 plus 51 times zero to the power of 50 plus one.

02:41 So let me write that out so you can see it.

02:43 So it would look like this.

02:47 101 times zero to the power of 100 plus 51 times zero to the power of 50 plus 1.

02:56 Now, all of that is going to be equal to just one because anything to the power of zero is just zero...

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