How do you use the Intermediate Value Theorem to show that the polynomial function #f(x) = x^4 + 8x^3 - x^2 + 2# has a zero in the interval [-1, 1]?

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Answer 1 · 2015-10-23T06:55:30

See the explanation.

#f(x)# is a polynomial and is continuous on #[-1,1]#.

#f(-1)=(-1)^4+8(-1)^3-(-1)^2+2=1-8-1+2=-6#

#f(1)=1^4+8*1^3-1^2+2=1+8-1+2=10#

By IVT, #f(x)# takes any value between #-6# and #10# on #[-1,1]# and hence: #f(c)=0, c in [-1,1]#.