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Critical Numbers of Asymptotes

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(1) Given function: f(x) = (x^2)*[e^(12x)]
(a) Find its horizontal asymptotes.
(b) Find its critical numbers.
(c) Find its inflection points.

(2) Given function: f(x) = (x^3) / [(x^2) - 16]
(a) Find its critical numbers.
(b) Find its inflection points.

This shows how to find critical numbers and inflection points of given functions. The critical numbers and inflection on points are determined.

Please see the attached file.

(1) Given function: f(x) = (x^2)*[e^(12x)]
(a) Find its horizontal asymptotes.

but
So, y=0 is the only horizontal asymptote

(b) Find its critical numbers.
Since , let f'(x)=0, we get
...

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"Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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