The Intermediate Value Theorem states that, if a continuous function over an interval #[a,b]# is such that #f(a)=x# and #f(b)=y#, then #f# assumes all possible values between #x# and #y# in the interval #[a,b]#. This function is mostly used to find zeroes of a function, by finding a point in which the function is negative, and one in which it's positive: passing from negative to positive (i.e., from below to above the #x#-axis), the function must cross it, and that's the root.
So, we must verify that #f(x)# is continuous in #[-3,-1]#, and that #f(-3)# and #f(-1)# have opposite signs.
Coefficients apart, the first two terms are power of #x#, and so they are continuous everywhere. The third term, #1/x#, has its only discontinuity in #x=0#, which is outside of our domain #[-3,-1]#, so #f# is indeed continuous in #[-3,-1]#.
Now we only need to compute:
#f(-3)=(-3)^3/2 - 4*(-3) + 1/-3 = -11/6<0#
#f(-1)= -1/2 -4(-1) -1 = 4-1-1/2 = 5/2>0#.
