Final answer:
To find where the function f(x) = e^{2x} e^{-x} is increasing or decreasing, we simplify it and find its derivative. f(x) simplifies to e^x, and since the derivative of e^x is also e^x and is always positive, f(x) is increasing on all real numbers.
Step-by-step explanation:
The problem requires us to analyze the behavior of the function f(x) = e^2x e^−x. To determine where the function is increasing or decreasing, we need to calculate the derivative of the function, f'(x), which will give us the rate of change. If f'(x) is positive on an interval, the function is increasing there; if f'(x) is negative, the function is decreasing.
First, let's simplify the function using properties of exponents:
f(x) = e^2x × e^−x = e(2x − x) = e^x
Now, find the derivative of f(x):
f'(x) = d/dx [e^x] = e^x
Since e^x is always positive, f'(x) is always positive. Therefore, the function is increasing on the interval ∀ x ∈ ℝ, where ℝ denotes the set of all real numbers.