Question

given the function $f(x) = x^4 + 6x^3 - x^2 - 30x + 4$, use the intermediate value theorem to decide which of the following intervals contains at least one zero. select all that apply. (2 points) $-5,-4$ $-4,-3$ $-3,-2$ $-1,0$ $0,1$ $1,2$

Explanation:

The Intermediate Value Theorem states that if a function \( f(x) \) 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 the interval \((a, b)\) such that \( f(c) = k \). In particular, if \( f(a) \) and \( f(b) \) have opposite signs, then there is at least one zero of the function in the interval \((a, b)\).

We are given the function \( f(x) = x^4 + 6x^3 - x^2 - 30x + 4 \). We need to evaluate \( f(x) \) at the endpoints of each interval and check if the function values have opposite signs.

Step 1: Evaluate \( f(x) \) at \( x = -5 \) and \( x = -4 \)

For \( x = -5 \):

$$
LATEXBLOCK0
$$

For \( x = -4 \):

$$
LATEXBLOCK1
$$

Since \( f(-5) = 4 \) and \( f(-4) = -20 \), and \( 4 \) and \( -20 \) have opposite signs, by the Intermediate Value Theorem, there is at least one zero in the interval \([-5, -4]\).

Step 2: Evaluate \( f(x) \) at \( x = -4 \) and \( x = -3 \)

We already know \( f(-4) = -20 \).
For \( x = -3 \):

$$
LATEXBLOCK2
$$

Since \( f(-4) = -20 \) and \( f(-3) = 4 \), and \( -20 \) and \( 4 \) have opposite signs, there is at least one zero in the interval \([-4, -3]\).

Step 3: Evaluate \( f(x) \) at \( x = -3 \) and \( x = -2 \)

We know \( f(-3) = 4 \).
For \( x = -2 \):

$$
LATEXBLOCK3
$$

Since \( f(-3) = 4 \) and \( f(-2) = 28 \), both are positive, so no zero in \([-3, -2]\).

Step 4: Evaluate \( f(x) \) at \( x = -1 \) and \( x = 0 \)

For \( x = -1 \):

$$
LATEXBLOCK4
$$

For \( x = 0 \):

$$ f(0) = 0^4 + 6(0)^3 - 0^2 - 30(0) + 4 = 4 $$

Both \( f(-1) = 28 \) and \( f(0) = 4 \) are positive, so no zero in \([-1, 0]\).

Step 5: Evaluate \( f(x) \) at \( x = 0 \) and \( x = 1 \)

We know \( f(0) = 4 \).
For \( x = 1 \):

$$
LATEXBLOCK5
$$

Since \( f(0) = 4 \) and \( f(1) = -20 \), they have opposite signs, so there is at least one zero in \([0, 1]\).

Step 6: Evaluate \( f(x) \) at \( x = 1 \) and \( x = 2 \)

We know \( f(1) = -20 \).
For \( x = 2 \):

$$
LATEXBLOCK6
$$

Since \( f(1) = -20 \) and \( f(2) = 4 \), they have opposite signs, so there is at least one zero in \([1, 2]\).

Answer:

The intervals that contain at least one zero are:

• \([-5, -4]\)
• \([-4, -3]\)
• \([0, 1]\)
• \([1, 2]\)