"Roots (Zeros) Show that the functions in Exercises 19–26...

Question

"Roots (Zeros) Show that the functions in Exercises 19–26 have exactly one zero

Accepted Answer

Determine if the function F of X equals 2X minus cosine squared, FX divided by 4, plus 3 has exactly one roop in the interval from negative infinity to infinity. Now, to solve this, let's first check the limits of our function. So, we'll check the behavior as our function goes to infinity and negative infinity. We have the limit, as X approaches infinity. Of F of X. Now let's take a look here. We have 2 X minus cosine squared of X divided by 4 + 3. Now, cosine squared of X divided by 4 oscillates between the values of 0 and 1. Now, 2 X goes to infinity. And the 3 can be attached to the 2 X. In other words, this limit. As approaches infinity goes to infinity. Because of 2 X plus 3. Let's also check the limit as it goes to negative infinity. We have a similar situation with the cosine squared term. Where that part oscillates between the values of 0 and 1. Ants We have 2 X plus 3. Which goes to negative infinity. So we can then say the limit, as X approaches negative infinity, will also be negative infinity. Because of 2 X plus 3. Now that we know our behavior, we can use the intermediate value theorem. So, by the intermediate value theorem, Because We transition at some point. From negative. It's a positive There exists At least One the route On the interval And we can tell this based on our limits, going from negative infinity to positive infinity. Now, to determine if there's exactly one route, we will check the derivative. If our function is strictly increasing or decreasing, we will only have one route. So, let's check our derivative. F of X Of our function. 2 X minus cosine squared. X divided by 4 + 3. F X Will be found using a chain rule. We first have the derivative of 2 X, which is just 2. We have 2 2 Kosan Of X divided by 4. Multiplied by the inside derivative. In this case, our interior derivative will be negatives of X divided by 4. All of this is multiplied by 1/4, based on the interior derivative. 4 x divided by 4. This gives us 2. Milla 2, cosine x divided by 4. Negative sign of X divided by 4. All divided by 4. We can now simplify this. Now we notice We actually have a double angle in the second. But first, we can simplify our 24ths, and our sons to get 2. Plus cosine x divided by 4. Sign of X divided by 4. All divided by 2 Using our Sine double angle formula, which tells us that sine to data. is equals to 2 Sign the, cosine theta. We end up getting 2. Plus Sign of x divided by 2. All divided by 4. Now we need to find the sign of this function. Our Of F of X This is going to be Always positive. This is because, while a sun does oscillate. 2 will always be greater than sun. Of X divided by 2, all divided by 4. This means that no matter what, since we're adding on to, this will always be a positive value. We can then confirm that F of X is always positive. Which means that FX It strictly increasing. From n infinity to infinity. We can then determine that by the intermediate value theorem, this has Exactly One Greeks. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

"Roots (Zeros) Show that the functions in Exercises 19–26 have exactly one zero

Key Concepts

Roots of a Function

Intermediate Value Theorem

Uniqueness of Zeros

Watch next