Question

Use the intermediate value theorem to show that there is a root of the given equation in the specified interval?

For x^4+x-3=0, (1,2)
can someone pleeeeaaase help me???

Answer 1

By applying the Intermediate Value Theorem to the continuous function f(x) = x^4 + x - 3, and showing that f(1) is negative and f(2) is positive, we confirm that there is at least one root in the interval (1, 2).

Step-by-step explanation:

The Intermediate Value Theorem states that for any continuous function f over an interval [a, b], if f(a) and f(b) have opposite signs, then there must be at least one root of the equation f(x) = 0 within that interval. To use the Intermediate Value Theorem for the equation x^4 + x - 3 = 0 within the interval (1, 2), we must first find the values of the function at x = 1 and x = 2.

Let's calculate:

• f(1) = 1^4 + 1 - 3 = 1 + 1 - 3 = -1 (which is less than 0)
• f(2) = 2^4 + 2 - 3 = 16 + 2 - 3 = 15 (which is greater than 0)

Since f(1) is negative and f(2) is positive, and the function f(x) = x^4 + x - 3 is continuous over the entire real line, by the Intermediate Value Theorem there must be at least one root between x = 1 and x = 2.