Theory and Examples

In Exerc...

Question

Theory and Examples

In Exercises 53 and 54, show that the function has neither an absolute minimum nor an absolute maximum on its natural domain.

y = x¹¹ + x³ + x − 5

Accepted Answer

Hi everyone. Let's take a look at this practice problem. This problem says determine if the function F of X, which is equal to x 7th, minus 2 x 5th plus 3X2 minus 7, has an absolute maximum or minimum overs domain. So the first thing we need to do is determine the domain of our function F of X. And if we look at our function F of X, we see that we have a polynomial, and recall that polynomials are defined for all real values of X. And so in our case, our domain is going to be the open interval from minus infinity to infinity. So now we have the domain determined, we can actually look at the end behavior of our function. And so we're gonna be looking at the case when X is approaching to infinity, and also when X approaches minus infinity. So, as X approaches infinity, we need to determine the behavior of F of X. So again, if we look at our function F of X, we see that the highest degree of our polynomial is 7, and so our function F of X is going to be dominated by the term X 7th. And so that means whatever the behavior is of X 7th, when X approaches infinity, that's going to be the same behavior as F of X. So, as X approaches infinity, X 7th also approaches infinity. Therefore, F of X is going to approach infinity. And if we look at the case, when X approaches minus infinity. Again, we're going to be looking for the behavior of F of X, and again, it's going to be dominated by the behavior of X 7th. And as X approaches minus infinity, X 7th also approaches minus infinity since we have an odd exponent on our power for X. And so that means that our function F of X is also going to approach minus infinity. So, if we look for the absolute. Maximum We'll see that there actually is no absolute maximum. That's because as X approaches infinity, our function F of X is unbound from above. It's also approaching infinity. So that means for our absolute maximum we actually have none. And if we look at the absolute minimum, We'll see a similar argument. We also have no absolute minimum because our function F of X is not bound by from below. So as x approaches minus infinity, our function F of X is also approaching minus infinity. So we don't really have an absolute minimum. So our absolute minimum is none. And so these two values, one for the absolute maximum and one for the absolute minimum. are the answers that this problem was looking for, and in both cases, we have none. So, I hope that this explanation has been useful, and I'll see you in the next video.

Theory and Examples

In Exercises 53 and 54, show that the function has neither an absolute minimum nor an absolute maximum on its natural domain.

y = x¹¹ + x³ + x − 5

Key Concepts

Natural Domain

Absolute Extrema

Behavior of Polynomial Functions

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