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Use the Intermediate Value Theorem to prove that every cubic function f(x)=Ax3+Bx2+Cx+Dhas at least one real root. You will have to first argue that you can find real numbers a and b so that f(a) is negative and f(b) is positive.

Short Answer

Expert verified

In either case, the intermediate value theorem applies to the continuous cubic function f(x)on [-N,N], and therefore f(x)has at least one real root on [-N,N).

Step by step solution

01

Step 1. Given Information.

Given expression f(x)=Ax3+Bx2+Cx+DAx3+Bx2+Cx+D

02

Step 2. The strategy is to prove that every cubic function stated above has at least one real root. 

If A>0then for large magnitude N we will have f(-N)<0<f(N)

And if A<0 then for large magnitude Nwe will have f(-N)<0<f(N)

In either case, the intermediate Value theorem applies to the continuous cubic function f(x)on[-N,N]and thereforef(x) has at least one real root on[-N,N]

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