Homework SourseLet fx xex For what values of x is the graph of f both decr
Let f(x) = xe^x.
For what values of x is the graph of f both decreasing and concave up? Explain your reasoning using algebra.
Please showcorrect work for full rating.
For what values of x is the graph of f both decreasing and concave up? Explain your reasoning using algebra.
Please showcorrect work for full rating.
Solution
No vertical asymptotes limf(x) x=>-infinity = limx e^x call e^x=z z=>0 and x= lnz so we have lim z*lnz with z=>0 = lim lnz/1/z =lim1/z/(-1/z^2)= lim-z =0 so y = 0 is a horizontal asymptote f´(x) = e^x(x+1)=0 x=-1 sign f´-------(-1)++++++ At x= -1 local minimumf(-1) = -1/e f´´(x) = e^x(x+1)+e^x = e^x(x+2) At x=-2 we have an inflection sign f´´ --------(-2 f(-2) =-2/e^2 -2>x concave downwards x>-2 concave upwards
