How do you use the intermediate value theorem to explain why #f(x)=x^3+3x-2# has a zero in the interval [0,1]?

1 Answer
Nov 19, 2016

Answer:

#f(x)# is continuous in the domain #x in [0,1]# and
#f(0) < 0 < f(1)#
therefore #EE_c : f(c) = 0# (by the intermediate value theorem)

Explanation:

The intermediate value says:

If #f(x)# is a continuous function over the domain #x in [a,b]#
then for any value #k in [f(a),f(b)]#
there is a value #c# such that #f(c)=k#

For the given function #f(0] = -2# and #f(1)=+2#
Since #k=0 in [-2,+2]#
there is a value #c# such that #f(c)=0#

[Technically, I have assumed without proof that #f(x)# is continuous within the given domain. If you need this proof, add as a new question or post a comment.]