application of Derivative

Question

find the largest interval on which the graph of the function f(x) = xe^x is concave downward?

Answer 1

The sign of the 2nd derivative indicates the direction of concavity.

 

f(x) = xe^x

 

f'(x)= xe^x + e^x

 

f"(x) = xe^{x +} e^{x +} e^x

 

f"(x) = e^x(x + 2)

 

Since ex is always positive, (x + 2) must be negative for the second derivative to be negative.

 

x + 2 < 0 where x , -2

 

f(x) is concave downward over the interval (–∞, –2)

 

A²