Question

Let f(x) = xe^x.

(a) Find the largest interval of the form [a, [infinity]) for which f is increasing.
(b) Restrict the domain of f to be the interval you found in part (a).

Answer 1

The largest interval on which f(x) = xe^x is increasing is [-1, infinity). This is found by taking the derivative of the function and determining where it is positive, which is for x > -1.

Step-by-step explanation:

To determine where the function f(x) = xe^x is increasing, we need to look at the derivative of the function. The derivative f'(x) tells us the slope of the function at any given point. So, we take the derivative:

f'(x) = e^x + xe^x (by applying the product rule).

Now, to find where the function is increasing, we find where f'(x) is positive:

e^x(1 + x) > 0.

Since e^x is always positive, we just need to determine when (1 + x) > 0, which is for x > -1.

Thus, the largest interval of the form [a, ∞) for which f is increasing is [-1, ∞). Restricting the domain of f to this interval means that we consider f only for x ≥ -1.