The function f(x) = (7 x+9)e^{-2 x} has one critical number. Find it.
Homework Posting Tips
Please show your work. Tutors will not do your homework for you. Please show your work for any question that you are posting.
To find the critical number of a function, we need to determine where the derivative of the function is equal to zero or does not exist.
First, let's find the derivative of the function f(x):
f'(x) = (7 + 7x - 18xe^(-2x))
Now, to find the critical number, we set the derivative equal to zero and solve for x:
(7 + 7x - 18xe^(-2x)) = 0
Since this equation is non-linear and involves the exponential function, it cannot be easily solved algebraically. We need to use numerical methods to approximate the solution.
One common numerical method is the Newton-Raphson method. Here's the step-by-step process using this method:
1. Make an initial guess for the critical number, let's say x = 0.
2. Apply the Newton-Raphson method to find a better approximation:
x_(n+1) = x_n - (f(x_n) / f'(x_n))
3. Repeat step 2 until the approximation converges to a solution. In this case, the solution will be the critical number.
Note: The Newton-Raphson method can be programmed into a calculator or implemented programmatically using a programming language like Python.
Alternatively, you can use online numerical solvers or graphing calculators that can find the critical number directly for you.