How do you use the Intermediate Value Theorem to show that the polynomial function #f(x) = 10x^4 - 2x^2 + 7x - 1# has a root in the interval [-3, 0]?

1 Answer
Oct 24, 2015

See the explanation.

Explanation:

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 #[-3,0]# where #f(c) = 0#

#f# is continuous on #[-3,0}# because it is a polynomial and polynomials are continuous at every real number.

#0# is between #f(-3) and #f(0)# because #f(-3) = 10(81)-2(9)+7(-3)-1# is positive and #f(0) = -1# is negative.

Therefore, by the Intermediate Value Theorem, there is a #c# in #[-3,0]# 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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