Question

determine the consecutive integer values of x between which each real zero of f(x)=-x^4 - x^3+4 is located by using a table. then sketch the graph on a separate sheet of paper.
a) zeros between x = -2 and x = 1, and x = -1 and x = 2
b) zeros between x = -2 and x = -1, and x = 1 and x = 2
c) zeros between x = -2 and x = 2, and x = 1 and x = -1
d) zero between x = -2 and x = 1

Explanation:

Step1: Evaluate function at integers

Let \(f(x)=-x^{4}-x^{3}+4\). Evaluate \(f(x)\) at \(x = - 2,-1,1,2\).
When \(x=-2\), \(f(-2)=-(-2)^{4}-(-2)^{3}+4=-16 + 8+4=-4\).
When \(x=-1\), \(f(-1)=-(-1)^{4}-(-1)^{3}+4=-1 + 1+4=4\).
When \(x = 1\), \(f(1)=-1^{4}-1^{3}+4=-1-1 + 4=2\).
When \(x = 2\), \(f(2)=-2^{4}-2^{3}+4=-16-8 + 4=-20\).

Step2: Use Intermediate - Value Theorem

Since \(f(-2)=-4\) and \(f(-1)=4\), by the Intermediate - Value Theorem (if a function \(y = f(x)\) 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 \(f(c)=k\)), there is a zero between \(x=-2\) and \(x = - 1\) because the sign of \(f(x)\) changes from negative to positive.
Also, since \(f(1)=2\) and \(f(2)=-20\), there is a zero between \(x = 1\) and \(x = 2\) because the sign of \(f(x)\) changes from positive to negative.

Answer:

B. zeros between \(x=-2\) and \(x=-1\), and \(x = 1\) and \(x = 2\)