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Problem 77 Hard Difficulty

Investigate the family of curves $ f(x) = e^x - cx $. In particular, find the limits as $ x \to \pm \infty $ and determine the values of $ c $ for which $ f $ has an absolute minimum. What happens to the minimum points as $ c $ increases?

Answer

no minimum

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Video Transcript

So we want to investigate the family of curves. Um And those curves are F of X equals E. To the X. In sc X. This is what we have here. And we're going to change the values for C. Like this. No. Um And we see That when C is equal to zero. Yes, it looks like the traditional cardiac function. When she is less than zero, we see that there isn't a minimum value um or when she is equal to zero, but when she is greater than zero, we see that there is a minimum value. As X goes to infinity, why goes to infinity. As X goes to negative infinity, why goes to infinity? So that would be the different ways of looking at it. And then when Y is negative as X goes negative infinity, why goes negative infinity? As X goes to positive infinity, Why goes to positive infinity? And then lastly when sequel, zero as X goes to infinity, why goes to zero And as X goes to positive infinity, why it goes to infinity.

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