For what interval is the function f(x)=3xe−x concave down?

Answer 1

f(x) = 3xe^-x

The graph is concave down when f"(x) < 0.

f'(x) = 3e^-x - 3xe^-x = 3e^-x(1 - x)

f"(x) = -3e^-x - 3e^-x(1 - x) = -3e^-x(2 - x) = 0 when x = 2.

If x < 2, then f"(x) < 0. So, the graph of y = f(x) is concave down.

If x > 2, then f"(x) > 0. So, the graph of y = f(x) is concave up.

The graph of y = f(x) is concave down on the interval (-∞, 2).