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

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Answer 1 · 2015-09-26T23:43:46

You can't. It does not have a zero in that interval.

It does have a zero in the interval #[0,5]#

#f# is continuous on #[0,5]#, because it is a polynomial function and all polynomial functions are continuous at every real number.

#f(0) = -2# and #f(5)=125-102=23#

#0# is between #f(0)# and #f(5)#.

Therefore, the Intermediate Value Theorem tells us that there is a number #c# in #[0,5]# with #f(c)=0#.