Absolute maxima and minima Determine the location and value of...

Question

Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.

ƒ(x) = sec x on [-(π/4),π/4]

Accepted Answer

Welcome back, everyone. In this problem, we want to find the absolute maximum and the minimum values of the function H on the interval open bracket, negative and a of pi, and Ace of pi closed bracket for H X equals the C count of 2 X. A says the absolute maximum is root 2 at X equals negative pi divided by 4 and pi divided by 8, and the absolute minimum is 1 at X equals 0. B says the maximum is root 2 at X equals negative pi divided by 8 and pi divided by 8, and the absolute minimum is 1 at X equals 0. C says the absolute maximum is 1 at X equals 0 and pi divided by 8, and the minimum is root 2 at X equals negative pi divided by 8 and pi divided by 8. And finally, it says the maximum is 1 at X equals 0 and X E equals pi divided by 8, while the minimum is root 2 at X E equals negative pi divided by 4 and pi divided by 8. Now, how are we going to figure out the maximum and minimum values? Well, the first thing we need to do is to find the derivative of H of X, OK? So now if we're differentiating H of X, OK. And we'll discuss why in a minute. We can differentiate the scant of 2 X by using the chain root. When we do that, we get the derivative of H X to be equal to the cant of 2 X. Multiplied by the tangent of 2 X. Multiplied by 2. Which we could rewrite as 26 2 X and 2 X. Now what are we supposed to do with our derivative? Well, recall that at the critical points, OK, at the critical points. The derivative, the first derivative is equal to 0. So if we can set our derivatives to zero, we'll be able to find the critical points and then we can use the critical points and the end points to help us find the absolute maximum and minimum values of each. So let's now go ahead and set our. First derivative to 0. So we can say that 22 X multiplied by the tangent of 2 X is going to be equal to 0. And if that's true, then that means it's the tangent value, the tangent of 2 X, which is gonna make it 0, OK? And if the tangent of 2 X equals 0, OK, then. This tells us that 2 X will be a multiple of pi, so we can say 2 X equals N pi. In that case, that means X is going to be equal to N pi. Divided by 2 where and Let's make that clear here is an integer. OK. So in other words, Ni is an integer of pi. Now, remember that these values vary for our range, or, or rather for our domain, sorry. But the interval we are focused on is between negative and 1/8 of pi, and then 1/8 of pi. So the question is which value. Of X or which value of N is going to be within this interval. Well, here, solving for negative and 8 of pi less than or equal to n pi divided by 2 less than or equal to. Pi divided by 8 and 8 of pi, then we find that the only value of N that satisfies this condition is 0, OK? Because 0. Multiplied by 1/2 of pie would be 0 and 0 is between negative and 1/8 of pi and 1/8 of pi. So therefore, This tells us that x equals 0 is the only critical point in this interval. So we have our critic at a point. Now that we have the critical point here, because here this is gonna be 0. Then we can evaluate H using the critical points and end points, OK? So evaluating H of X at critical points and endpoints. Means we're trying to find the value of H when X is 0. When X is negative pi divided by 8 and when X is pi divided by 8. Now when X is 0, H equals the 2 of 0, which equals 1. When X is negative pi divided by 8, H is thecant of negative pi divided by 8 multiplied by 2, thus negative pi divided by 4, and that would be equal to root 2. And when X equals pi divided by 8, H will be equal to the C of pi divided by 4, which is also equal to root 2. So what does this tell us? Well, here, notice that we have our maximum value, OK, at X equals negative pi divided by 8 and, OK, at X equals pi divided by 8. And we have our absolute minimum value, OK, at X equals 0. That's the absolute, the absolute maximum is root 2 at X equals negative pi divided by 8 and the pi divided by 8. And the absolute minimum is 1 at X equals 0. If we look back at our answer choices, that tells us B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.

ƒ(x) = sec x on [-(π/4),π/4]

Key Concepts

Absolute Extrema

Critical Points

Secant Function

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