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]?

Question

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]?

Christopher Allers · Accepted Answer

See the 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". .

Emma Aldridge · Answer

undefined To 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], you need to show that the function changes sign over that interval.

First, evaluate $ f(-3) $ and $ f(0) $. 
- $ f(-3) = 10(-3)^4 - 2(-3)^2 + 7(-3) - 1 $
- $ f(0) = 10(0)^4 - 2(0)^2 + 7(0) - 1 $

Then, determine the signs of $ f(-3) $ and $ f(0) $. If they are different, then by the Intermediate Value Theorem, there exists at least one root of the polynomial function in the interval [-3, 0].