Use the intermediate value theorem to show that there is a root of the equation x⁵ - 2x⁴ - x - 3 = 0 in the interval (2,3).

Solution:

Given, the equation is x⁵ - 2x⁴ - x - 3 = 0

We have to find the root of the equation in the interval (2, 3) using the intermediate value theorem.

Let f(x) = x⁵ - 2x⁴ - x - 3

f(2) = (2)⁵ - 2(2)⁴ - 2 - 3

f(2) = 32 - 32 - 5

f(2) = -5

f(3) = (3)⁵ - 2(3)⁴ - 3 - 3

f(3) = 243 - 162 - 6

f(3) = 75

Intermediate value theorem states that if a continuous function f(x) on the interval [a,b] has values of opposite sign inside an interval, then there must be some value x = c on the interval (a,b) for which f(c) = 0.

f(x) is continuous on the interval [2, 3] because it is a polynomial function, and is continuous at each point in the interval.

Here, f(2) is negative and f(3) is positive. 

Therefore, f(x) is continuous on the closed interval [2, 3], there must be some value x = c on the interval [2, 3] for which f(c) = 0.

Use the intermediate value theorem to show that there is a root of the equation x⁵ - 2x⁴ - x - 3 = 0 in the interval (2,3).

Summary:

Using the intermediate value theorem, the equation x⁵ - 2x⁴ - x - 3 = 0 is continuous on the closed interval [2, 3], there is a root of the equation in the interval (2,3).