Find the intervals where the function is increasing or decreasing.
y=e^(1-e^x).
Use the interval [0, 2pi]

Question

Find the intervals where the function is increasing or decreasing. 
y=e^(1-e^x). 
Use the interval [0, 2pi]

anonymous · Answer

What level math class are you in?

anonymous · Answer

Calculus

anonymous · Answer

Cool!

Do you know how to take the derivative of$$y=e^{1-e^{x}}$$?

anonymous · Answer

Yea is it e^(1-e^x)(-e^x)?

anonymous · Answer

Yes.

If you plot that function over the interval, it will give you a graph of the instantaneous slope of your original function at each point.

So wherever your derivative is positive, your original function is increasing.  Wherever it is negative, your original function is decreasing.  And if it is 0, then it is doing neither.

anonymous · Answer

I'm not really sure how exactly to do that... Would I have to find the critical numbers?

anonymous · Answer

You can either use a graphing calculator (if allowed by your class), or set the derivative equal to 0 and solve.  Then have your answers be intervals.  So if a, b, c, and d are your answers for example, make the intervals (a,b), (b, c), (c, d) and plug in a value of x that falls within each interval to see if the derivative is positive or negative there.  Then use what I said above the get your intervals for either increasing or decreasing.

anonymous · Answer

Ok got it, but whenever I put the derivative equal to 0 I get an undefined number. Is that how it is or am I doing it wrong?

anonymous · Answer

You're not doing it wrong.  That means that the derivative never equals zero, so the function is always either increasing or decreasing.

anonymous · Answer

Oh ok so I used pi. And the answer was negative once I put it into the derivative so that means within that interval it is decreasing?

anonymous · Answer

Yep

anonymous · Answer

Ok thank you so much :)

anonymous · Answer

You're welcome.