Let f be the function f(x)=2xe^x. The graph of f is concave down when a. x-2 c. x-1 e. x

Question

Let f be the function f(x)=2xe^x. The graph of f is concave down when a. x<-2 b.x >-2 c. x<-1 d. x>-1 e. x<0

kinggeorge · Answer

First, you need to take the second derivative. It's a little bit of work, but you get the equation $$2 e^x (2+x)$$Now we want this to be less than 0.

anonymous · Answer

can you showme howto do the whole problem?

anonymous · Answer

please

anonymous · Answer

x<-2

anonymous · Answer

$$f(x)=2xe^x$$ product rule 
$$f'(x)=2xe^x+2e^x=2(xe^x+e^x)$$ second derivative product rule again 
$$f''(x)=2(xe^x+e^x+e^x)=2(xe^x+2e^x)=2e^x(x+2)$$

anonymous · Answer

you want to know where the second derivative is negative. $e^x>0$ so we can ignore that factor and solve $x+2<0$ making $x<2$
that is where the second derivative if negative, and so that is where the function is concave  down

anonymous · Answer

oops!

anonymous · Answer

$$x+2<0\implies x<-2$$ must be getting late

kinggeorge · Answer

Had to step away for a bit. Sorry about that. 

But now we need to notice that $2e^x$ is always greater than 0. Thus, if we want the second derivative must be negative, we do what satellite just did above me. So $$x<-2$$