The interval in which y=x2e−x𝑦=𝑥2𝑒-𝑥 is increasing is

To find the interval in which the function y = x^2 * e^-x is increasing, we need to find its derivative and set it equal to zero to find the critical points.

Step 1: Find the derivative of the function. The derivative of y = x^2 * e^-x using the product rule and the chain rule is y' = 2x * e^-x - x^2 * e^-x.

Step 2: Set the derivative equal to zero and solve for x. 0 = 2x * e^-x - x^2 * e^-x 0 = e^-x * (2x - x^2) Setting each factor equal to zero gives x = 0 and x = 2 as critical points.

Step 3: Test the intervals determined by the critical points. Choose test points in the intervals (-∞, 0), (0, 2), and (2, ∞) and substitute them into the derivative to determine whether the function is increasing or decreasing in each interval.

For (-∞, 0), choose x = -1. Substituting -1 into the derivative gives a positive value, so the function is increasing on (-∞, 0).

For (0, 2), choose x = 1. Substituting 1 into the derivative gives a positive value, so the function is increasing on (0, 2).

For (2, ∞), choose x = 3. Substituting 3 into the derivative gives a negative value, so the function is decreasing on (2, ∞).

Therefore, the function y = x^2 * e^-x is increasing on the intervals (-∞, 0) and (0, 2).