Given the function f(x)=(4x+8)e-x.Determine where the function is increasing and where itis decreasing.Determine where the graph of the function fis concave up and where itis concave down.Find relative extremum of the function f(x).Find absolute extremum of the function f(x)on-4,0.Solution1We have f'(x)==0 therefore x=Basing on the table of sign off'(x)onRwe conclude that f(x) increases asxand decreases asx12We have f''(x)==0 therefore Basing on the table of sign off''(x)onRwe conclude that the graph of the function is concave down asxand is concave upasx3Moreover, we see that f(x) has aatx=and it has a global minimum atx=15Total area of the model isS(r,h)=r,hV(r,h)=r,hS=100*f(0)we express h respect tor and substitute h into the volume function V(r)='write the expression respect torV(r)on its domain r=0we see that the model will achieve the largest volume when r=meter (round the result with 1 digit)