It seems you've uploaded an image related to applying the Intermediate Value Theorem. The problem is asking to show that there is a root of the equation $x^4 + x - 5 = 0$ within the interval (1, 2).

Let's break this down step by step:

1. Function Definition: The function given is $f(x) = x^4 + x - 5$.

2. Continuity: The function $f(x)$ is continuous on the interval $[1, 2]$ because it is a polynomial, and polynomials are continuous over all real numbers.

3. Evaluate $f(1)$ and $f(2)$:

   • $f(1) = 1^4 + 1 - 5 = 1 + 1 - 5 = -3$
   • $f(2) = 2^4 + 2 - 5 = 16 + 2 - 5 = 13$

4. Intermediate Value Theorem Application:

   • The Intermediate Value Theorem states that if a function is continuous on an interval $[a, b]$ and takes values of opposite signs at the endpoints of the interval, then there must be at least one $c$ in the interval $(a, b)$ such that $f(c) = 0$.
   • In this case, $f(1) = -3$ and $f(2) = 13$, which are of opposite signs. Therefore, by the Intermediate Value Theorem, there exists a value $c$ in $(1, 2)$ such that $f(c) = 0$, meaning there is a root in the interval $(1, 2)$.

Filling in the blanks:

• $f(x) = x^4 + x - 5$ is continuous on the closed interval $[1, 2]$, and $f(1) = -3$, $f(2) = 13$.
• Since $f(1) = -3$ and $f(2) = 13$, there is a number $c$ in $(1, 2)$ such that $f(c) = 0$ by the Intermediate Value Theorem. Thus, there is a root of the equation $x^4 + x - 5 = 0$ in the interval $(1, 2)$.

Would you like more details or help with another concept?

Here are some additional related questions:

1. How does the Intermediate Value Theorem apply to other functions?
2. What are some examples of continuous functions where we can use the theorem?
3. How do we graphically interpret the result of the Intermediate Value Theorem?
4. How do we find the approximate value of the root in the interval?
5. Can the Intermediate Value Theorem be used for non-continuous functions?

Tip: Always check that the function is continuous on the interval before applying the Intermediate Value Theorem!