# Solve "Derivative Problem: Find Min & Max of (e^(-x)) - (e^(-2x))

• Eshi
In summary, the conversation discusses finding the minimum and maximum values of a function using derivatives. The attempt at a solution involves finding the derivative and setting it equal to zero. However, there was a sign error in the derivative calculation which was corrected. The final solution involves dividing both sides by the exponential term and simplifying to find the values of x that make the derivative equal to zero.

## Homework Statement

I have to find the min and max of this function using derivatives:

(e^(-x)) - (e^(-2x))

## The Attempt at a Solution

f'(x) = -e^(-x) + 2x(e^(-2x))
So now i set that to zero, and I get...

2x(e^(-2x)) = e^(-x)

And at this point I have no idea what to do. If you divide e^-x by e^-2x can u do something with the exponents?

Double check your derivative, you shouldn't be bringing an x down right? The derivative of ecx for a constant c is just cecx

o i ggot it! so it become

2 = e^(-x-2x)
and then you simply take the ln...

thanks jeffreydk

No problem.

Watch out though, I think you have a sign error in there.

f'(x)=-e-x+2e-2x=0

So then 2e-2x=e-x

and therefore by dividing you get 2=e-x+2x

## 1. What is a derivative?

A derivative is a mathematical concept that measures the rate of change of a function at a specific point. It tells us how the output of a function changes when the input changes.

## 2. How do you find the derivative of a function?

To find the derivative of a function, we use the rules of differentiation. These include the power rule, product rule, quotient rule, and chain rule. We also need to know the basic derivatives of common functions such as polynomials, exponentials, and trigonometric functions.

## 3. What is the purpose of finding the minimum and maximum of a function?

Finding the minimum and maximum of a function helps us understand the behavior of the function and identify important points, such as the highest or lowest points, on the graph. It can also help us optimize a function to find the best solution to a problem.

## 4. How do you find the minimum and maximum of a function?

To find the minimum and maximum of a function, we first take the derivative of the function and set it equal to 0. Then, we solve for the variable to find the critical points. We can then plug these critical points back into the original function to determine the minimum and maximum values.

## 5. Can you explain the process of solving the derivative problem for (e^(-x)) - (e^(-2x))?

Sure! First, we take the derivative of the function using the power rule and chain rule. This gives us the equation -e^(-x) + 2e^(-2x). Next, we set this equal to 0 and solve for x, which gives us x = ln(2). Plugging this value back into the original function, we get a minimum value of -1/2 at x = ln(2) and a maximum value of -1 at x = 0.