Question

Use the intermediate value theorem to show that the polynomial has a real zero between the given integees. f(x) = x ^ 3 - 4x - 3 between 1 and 7

Answer 1

$f(x)=x^3-4x-3$

The Intermediate Value Theorem states that if f(a) and f(b) have opposite signs, then there exists at least one value c between a and b for which f(c)=0.

Therefore, for the interval:

$x\in\lbrack1,7\rbrack$$\begin{gathered} a=1 \\ b=7 \\ so\colon \\ f(a)=f(1)=1^3-4(1)-3=1-4-3=1-7=-6 \\ f(b)=f(7)=(7)^3-4(7)-3=343-28-3=312 \end{gathered}$

Since:

$\begin{gathered} f(a)<0 \\ f(b)>0 \end{gathered}$

by the Intermediate Value Theorem, there must be at least one real zero between 1 and 7.