Question

Determine the consecutive integer values of x between which each real zero is located.

f
(
x
)
=
x
3
−
3
x
2
+
2
f(x)=x
3
−3x
2
+2

Answer 1

To determine the consecutive integer values of \( x \) between which each real zero of the function \( f(x) = x^3 - 3x^2 + 2 \) is located, we can use the Intermediate Value Theorem. This theorem states that if a continuous function changes sign over an interval \([a, b]\), then there is at least one root in that interval.

First, we will evaluate \( f(x) \) at a few integer values to find where the function changes sign.

1. Calculate \( f(0) \):
\[
f(0) = 0^3 - 3 \cdot 0^2 + 2 = 2
\]

2. Calculate \( f(1) \):
\[
f(1) = 1^3 - 3 \cdot 1^2 + 2 = 1 - 3 + 2 = 0
\]

3. Calculate \( f(2) \):
\[
f(2) = 2^3 - 3 \cdot 2^2 + 2 = 8 - 12 + 2 = -2
\]

4. Calculate \( f(3) \):
\[
f(3) = 3^3 - 3 \cdot 3^2 + 2 = 27 - 27 + 2 = 2
\]

Now we can compile our results:
- \( f(0) = 2 \)
- \( f(1) = 0 \)
- \( f(2) = -2 \)
- \( f(3) = 2 \)

From the calculations:
- \( f(0) \) and \( f(1) \): \( f(0) > 0 \) and \( f(1) = 0 \) → one zero is located at \( x = 1 \).
- \( f(1) \) and \( f(2) \): \( f(1) = 0 \) and \( f(2) < 0 \) → there is a root at \( x = 1 \) and another root located between \( 1 \) and \( 2 \) (since \( f(2) < 0 \)).
- \( f(2) \) and \( f(3) \): \( f(2) < 0 \) and \( f(3) > 0 \) → this indicates there is another real zero between \( 2 \) and \( 3 \).

Thus, we conclude:
- There is a root at \( x = 1 \).
- There is another root between \( 1 \) and \( 2 \).
- There is yet another root between \( 2 \) and \( 3 \).

In summary, the intervals containing the real zeros of the function are:
- A root at \( x = 1 \) (exactly).
- A root located in the interval \( (1, 2) \).
- A root located in the interval \( (2, 3) \).