Question

determine the consecutive integer values of x between which each real zero of $f(x) = -x^3 + 2x^2 - 4$ is located by using a table.

a) zero between $x = 0$ and $x = 1$

b) zeros between $x = -2$ and $x = -1$, and $x = 0$ and $x = -4$

c) zero between $x = 0$ and $x = -1$

d) zero between $x = -2$ and $x = -1$

Explanation:

Step1: Evaluate f(x) at x=-2

$f(-2) = -(-2)^3 + 2(-2)^2 - 4 = 8 + 8 - 4 = 12$

Step2: Evaluate f(x) at x=-1

$f(-1) = -(-1)^3 + 2(-1)^2 - 4 = 1 + 2 - 4 = -1$

Step3: Evaluate f(x) at x=0

$f(0) = -(0)^3 + 2(0)^2 - 4 = 0 + 0 - 4 = -4$

Step4: Evaluate f(x) at x=1

$f(1) = -(1)^3 + 2(1)^2 - 4 = -1 + 2 - 4 = -3$

Step5: Analyze sign changes

A sign change between $f(a)$ and $f(b)$ means a zero between $a$ and $b$. $f(-2)=12$ (positive) and $f(-1)=-1$ (negative) have a sign change. No sign changes occur for other consecutive integer pairs.

Answer:

D) zero between $x = -2$ and $x = -1$