Question

if f(x) = x^3 - 4x + 6, show that there are values c for which f(c) equals (a) $pi$, (b) - $sqrt{3}$, and (c) 5,000,000. so, continue using larger values of x. find f(175). f(175)=5358681 (simplify your answer.) does a solution exist between 170 and 175 for f(x)=5,000,000? yes, because f(170)<5,000,000<f(175). inconclusive, because 5,000,000 does not lie between f(170) and f(175). inconclusive, because f(170)<0 but f(175)>0. yes, because f(170)>5,000,000>f(175).

Explanation:

Step1: Recall Intermediate - Value Theorem

The Intermediate - Value Theorem states that if \(y = f(x)\) is continuous on a closed interval \([a,b]\), and \(k\) is a number between \(f(a)\) and \(f(b)\), then there exists at least one number \(c\) in \((a,b)\) such that \(f(c)=k\). The function \(f(x)=x^{3}-4x + 6\) is a polynomial, so it is continuous everywhere.

Step2: Analyze part (c)

We know \(f(170)=170^{3}-4\times170 + 6=4913000-680 + 6=4912326\) and \(f(175)=5358681\). Since \(4912326<5000000<5358681\), by the Intermediate - Value Theorem, there exists a number \(c\in(170,175)\) such that \(f(c) = 5000000\).

Answer:

A. Yes, because \(f(170)<5000000<f(175)\)