Intermediate Value Theorem

Question

1. Use the Intermediate Value Theorem to show that there exists at least one solution in the set of real numbers for the following equation. (Note that you are NOT provided with any interval).

x⁵ − x³ + 3x = 5

Answer 1

Note that y=-5 at x=0 and 25 at x=2 and the function is continuous (polynomial) thus guaranteeing at least one solution betweeon x=0 and x=2.

Answer 2

Let f(x) = x⁵ - x³ + 3x.

The intermediate value theorem just says that if f(x) < 5 somewhere and f(x) > 5 somewhere else, then there has to be a point where f(x) = 5 as well.

We can see easily that f(x) = 0 when x=0 and f(x) approaches infinity when x is large, so somewhere between x=0 and positive infinity, f(x) must equal 5.