For R"(x) = -15[(x-1)(e^-x)-(e^-x)] with 0 < x < 7

What interval is the graph concave and and concave down?

I know concave up is (0,2). Is that right? Does concave down exist?

You need to factor out the e^-x as well, giving you

R" = -15e^-x (x-2)
R" >0 for x<2

You are correct. Since R" < 0 for x>2, the graph is concave down on (2,7)