What's the practical domain and range of this volume function: 4x^3-39x^2+93.5x

Answer 1

for large |x| the function is dominated by x^3
so
for a large value of x the function is large positive
for a large negative x the function is large negative
so
the range is -oo to +oo
HOWEVER if it describes a volume, it must be +
where is it + ?
find the zeros
v = x (4x^2 -39 x + 93.5)
0 at x = 0 of course
also at x = [ 39 +/- 5 ] / 8
or x = 5.5 and 4.25

so the function comes up from -oo to x = 0, then is positive from x = 0 to x = 4.25, then is negative from x = 4.25 to x = 5.5. Then it heads off to the upper right. Hope that helps.