Increasing and decreasing functions. Find the intervals on whi...

Question

Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.

f(x) = eˣ/(e²ᵉ + 1)

Accepted Answer

Welcome back, everyone. Find the intervals where the function G of X equals 3E to the power of X divided by E to the power of 2 X plus 1 is increasing and decreasing. So first of all, let's consider the interval where the function is increasing. If G of X is increasing, this means that its first derivative must be positive. So we have to identify the first derivative. E X and we are going to apply the quotient rule which says that first of all we take the derivative of the top function, so the derivative of 3 e to the power of X. Multiply it by the bottom function which is e to the power of 2 x plus 1. We then subtract the top function 3 is the power of X, multiplied by the derivative of the bottom function E to the power of 2 X plus 1. And we have to divide all of that by the square of the bottom function. It's the power of 2 x plus 1 squared. So now let's evaluate the results. We get 3 it's the power of X for the derivative. Multiplied by e to the power of 2 x plus 1. Now what is the derivative of E to the power of 2 X? To begin with, we get 3 e's the power of x multiplied by E to the power of 2 X, right, because the derivative of e to the power of X is eat the power of X, but here we have to apply the chain rule. So we also multiply by the derivative of 2 X because our exponent is a function and the derivative of 1 is 0. Then we divide everything by e to the power of 2 X +12. Let's recall that the derivative of 2 axis simply 2 because the derivative of X is 1. And now we can start simplifying. So we get 3 E to the power of X, multiplied by E to the power of 2 X plus 1. Minus 2 multiplied by 3 gives us 6, E to the power of X, Multiplied by e to the power of 2 X. All of that Is divided by the power of 2 X +12. And now we can factor out 3. It's the power of X. Which leaves us with it's the power of 2 X plus 1. Minus 2 E to the power of 2 X. All of that Divided by each the power of 2 x plus 1 square. Simplifying further, we get 3 E to the power of X, multiplied by 1 minus E to the power of 2 X. Divided by Eats the power of 2 x plus 1 square. So now, when is this positive? Well, we have to recall that the exponential terms eats the power of X. And e is the power of 2 X. They are always positive no matter what X is. Our denominator is always positive because it's a square of a term. E to the power of 2 x is always positive, and we're adding 1, so it's always positive. So we only want to focus on the numerator. G of X. is positive if the numerator is positive because the denominator is positive in all the domain right for any X. So we want to understand that we are interested in 1 minus e to the power of 2 X being positive. Why are we ignoring 3 E's the power of X? Well, because it's also always positive. So basically we can. Label the terms that are always positive and only focus on the remaining factor 1 minus eats the power of two Xs. It must be positive. To make sure that the function is increasing. So this means that E to the power of 2 X is less than 1, right? Now, let's recall that it's the power of 2 X is equal to 1 if X is equal to 0. So 0 is the point of interest. Specifically, we can visualize this exponential function if we want. For complete clarity. Or negative values of X. Each to the power of X is less than 1. So if X is equal to 0, each of the power of X is 1, and for any X, less than 0 eats the power of X will be less than 1. So if we want to get the power of X, the power of 2 X less than 1. This means that X must be less than 0. Now, this is where the function is increasing, and now let's identify G of X being less than 0. So this means that 1 minus E to the power of 2 X must be less than 0. And eats the power of 2 X. Must be greater than one in this case. So if it's greater than 1, this means that X is greater than 0. Well then, now, what we want to do is simply summarize our findings. So using all of this information, we can say that the function is increasing where the derivative is positive. X is less than 0, so X. Belongs to the interval from negative infinity up to 0. And now the function is decreasing where the derivative is negative. So X must be greater than 0, so from 0 up to infinity. And those will be our final answers to this problem. Thank you for watching.

Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.f(x) = eˣ/(e²ᵉ + 1)

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Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.

f(x) = eˣ/(e²ᵉ + 1)