Questions: Explain why the function has at least one zero in the given interval. Function Interval f(x) = (1/13) x^4 - x^3 + 6 [1,2] At least one zero exists because f(x) is continuous and f(1)<0 while f(2)<0. At least one zero exists because f(x) is continuous and f(1)>0 while f(2)<0. At least one zero exists because f(x) is continuous and f(1)>0 while f(2)>0. At least one zero exists because f(x) is not continuous. At least one zero exists because f(x) being a fourth degree polynomial must have a real solution.

Explain why the function has at least one zero in the given interval.
Function Interval
f(x) = (1/13) x^4 - x^3 + 6 [1,2]
At least one zero exists because f(x) is continuous and f(1)<0 while f(2)<0.
At least one zero exists because f(x) is continuous and f(1)>0 while f(2)<0.
At least one zero exists because f(x) is continuous and f(1)>0 while f(2)>0.
At least one zero exists because f(x) is not continuous.
At least one zero exists because f(x) being a fourth degree polynomial must have a real solution.

Transcript text: Explain why the function has at least one zero in the given interval. Function Interval \[ f(x)=\frac{1}{13} x^{4}-x^{3}+6 \quad[1,2] \] At least one zero exists because $f(x)$ is continuous and $f(1)<0$ while $f(2)<0$. At least one zero exists because $f(x)$ is continuous and $f(1)>0$ while $f(2)<0$. At least one zero exists because $f(x)$ is continuous and $f(1)>0$ while $f(2)>0$. At least one zero exists because $f(x)$ is not continuous. At least one zero exists because $f(x)$ being a purth degree polynomial must have a real solution.

Solution

Solution Steps

To determine if the function $ f(x) = \frac{1}{13} x^{4} - x^{3} + 6 $ has at least one zero in the interval $[1, 2]$, we can use the Intermediate Value Theorem. This theorem states that if a function is continuous on a closed interval $[a, b]$ and takes on opposite signs at the endpoints of the interval, then there is at least one zero in the interval. We need to check the values of $ f(x) $ at $ x = 1 $ and $ x = 2 $.

Solution Approach

Evaluate $ f(x) $ at $ x = 1 $ and $ x = 2 $.
Check the signs of $ f(1) $ and $ f(2) $.
If $ f(1) $ and $ f(2) $ have opposite signs, then by the Intermediate Value Theorem, there is at least one zero in the interval $[1, 2]$.

Step 1: Evaluate the Function at the Endpoints

First, we evaluate the function $ f(x) = \frac{1}{13} x^{4} - x^{3} + 6 $ at the endpoints of the interval $[1, 2]$.

\[ f(1) = \frac{1}{13} \cdot 1^{4} - 1^{3} + 6 = \frac{1}{13} - 1 + 6 \approx 5.0769 \]

\[ f(2) = \frac{1}{13} \cdot 2^{4} - 2^{3} + 6 = \frac{16}{13} - 8 + 6 \approx -0.7692 \]

Step 2: Check the Signs of the Function Values

Next, we check the signs of $ f(1) $ and $ f(2) $:

\[ f(1) \approx 5.0769 \quad (\text{positive}) \]

\[ f(2) \approx -0.7692 \quad (\text{negative}) \]

Step 3: Apply the Intermediate Value Theorem

Since $ f(x) $ is a polynomial, it is continuous on the interval $[1, 2]$. The Intermediate Value Theorem states that if a continuous function takes on opposite signs at the endpoints of an interval, then there is at least one zero in that interval.

Given that $ f(1) > 0 $ and $ f(2) < 0 $, there must be at least one zero in the interval $[1, 2]$.