To identify the intervals of length [tex]\(\frac{1}{4}\)[/tex] which contain a root of the function [tex]\(f(x) = x^7 + 3x - 13\)[/tex], we need to analyze the given intervals and determine where a root is likely to be present based on the information provided.
The function has been given as [tex]\(f(x) = x^7 + 3x - 13\)[/tex].
To find the intervals that contain a root, it is necessary to verify if [tex]\(f(x)\)[/tex] changes sign in those intervals. If [tex]\(f(a) \cdot f(b) < 0\)[/tex] for any interval [tex]\([a, b]\)[/tex], then by the Intermediate Value Theorem, there is at least one root of [tex]\(f(x)\)[/tex] in the interval [tex]\([a, b]\)[/tex].
The intervals under consideration are:
1. [tex]\([1.40, 1.65]\)[/tex]
2. [tex]\([1.10, 1.35]\)[/tex]
3. [tex]\([1.35, 1.60]\)[/tex]
4. [tex]\([1.25, 1.50]\)[/tex]
5. [tex]\([1.30, 1.55]\)[/tex]
Based on our analysis, the intervals that contain the roots of the function [tex]\(f(x)\)[/tex] are:
- [tex]\([1.35, 1.60]\)[/tex]
- [tex]\([1.25, 1.50]\)[/tex]
- [tex]\([1.30, 1.55]\)[/tex]
Thus, the intervals of length [tex]\(\frac{1}{4}\)[/tex] which contain a root of the function [tex]\(f(x) = x^7 + 3x - 13\)[/tex] are:
- [tex]\([1.35, 1.60]\)[/tex]
- [tex]\([1.25, 1.50]\)[/tex]
- [tex]\([1.30, 1.55]\)[/tex]
In conclusion, the roots of the function are located within the specified intervals above.
