How do you use the Intermediate Value Theorem to show that the polynomial function #f(x) = x^3 + 2x - 1# has a zero in the interval [0,1]?

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Answer 1 · 2015-10-26T14:34:47

See the explanation section below.

The Intermediate Value Theorem says that

if #f# is continuous on #[a,b]#, then #f# attains every value between #f(a)# and #f(b)# at #x# values in #[a,b]#

In general, to show that a function attains a chosen value, call it #N#, on interval #[a,b}# using the IVT requires:

Show that the function is continuous on #[a,b]#

Show that #N# is between #f(a)# and #f(b)#

Conclude, using the Intermediate Value Theorem, that there is a #c# in #[a,b]# with #f(c) = N#

In this case, #f(x)# has a root #c# if and only if #f(c) = 0#, so we need to show that there is a #c# in #[0,1]# where #f(c) = 0#

#f# is continuous on #[0,1]# because it is a polynomial and polynomials are continuous at every real number.

#0# is between #f(0) and #f(1)# because #f(0) = -1# is negative and #f(1) = 2# is positive.

Therefore, by the Intermediate Value Theorem, there is a #c# in #[0,1]# with #f(c) = 0#.

This #c# is a root of #f(x)#.

Final Note In mathematics "there is a" mean "there is at least one". It does NOT mean "there is exactly one".
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