Use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval.

Question

Use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval.

f(x) = x⁵ − 3x ,between x = 1 and x = 2

Substitute x = 1 and x = 2 into the function and simplify. 

f(1)=

f(2)=

Interpret the results using the Intermediate Value Theorem.

Because f is a polynomial function and since f(1) is  negative/positive and f(2) is negative/positive , there is at least one real zero between x = 1and x = 2

Answer 1

f(1) = 1^5 - 3(1) = 1-3 = -2 <0

f(2) = 2^5 - 3(2) = 32-6 = 26 >0

IVT states that there must be a value where f(x) = 0, as this is a continuous polynomial

function on the interval

the approximate solution is by the way 1.316 per the graph