Q. 87 - Calculus

Question

Use the Intermediate Value Theorem to prove that every cubic function $f\left(x\right)=A{x}^3+B{x}^2+Cx+D$ has 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.

Step-by-Step Solution

Answer

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

Step 1. Given Information.

Given expression  $f\left(x\right)=A{x}^3+B{x}^2+Cx+D$$A{x}^3+B{x}^2+Cx+D$

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

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

And if $A<0$  then for large magnitude $N$ we will have  $f(-N)<0<f\left(N\right)$

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