If #f(x)=x^5-2x^4-x-3#
then
#color(white)("XXX")f(color(blue)2)=color(blue)2^5-2 * color(blue)2^4-color(blue)2-3=color(red)(-5)#
and
#color(white)("XXX")f(color(blue)3)=color(blue)3^5-2 * color(blue)3^4-color(blue)3-3=243-162-3-3=color(red)(+75)#
Since #f(x)# is a standard polynomial function, it is continuous.
Therefore, based on the intermediate value theorem, for any value, #color(magenta)k#, between #color(red)(-5)# and #color(red)(+75)#, there exists some #color(lime)(hatx)# between #color(blue)2# and #color(blue)3# for which #f(color(lime)(hatx))=color(magenta)k#
Since #color(magenta)0# is such a value,
there exists some value #color(lime)(hatx) in [color(blue)2,color(blue)3]# such that #f(color(lime)(hatx))=color(magenta)0#
