Question

use the intermediate value theorem to confirm that the given polynomial has at least one zero within the given interval: (f(x)=x^{5}-5x), 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 select an answer and (f(2)) is select an answer, there is at least one real zero between (x = 1) and (x = 2). question help: message instructor

Explanation:

Step1: Calculate f(1)

Substitute $x = 1$ into $f(x)=x^{5}-5x$:
$f(1)=1^{5}-5\times1=1 - 5=-4$

Step2: Calculate f(2)

Substitute $x = 2$ into $f(x)=x^{5}-5x$:
$f(2)=2^{5}-5\times2=32 - 10 = 22$

Step3: Apply Intermediate - Value Theorem

Since $f(x)$ is a polynomial, it is continuous everywhere. $f(1)=-4<0$ and $f(2)=22>0$. By the Intermediate - Value Theorem, for any value $y = 0$ (since we are looking for zeros) between $f(1)$ and $f(2)$, there exists at least one $c\in(1,2)$ such that $f(c)=0$.

Answer:

$f(1)=-4$
$f(2)=22$
Because $f$ is a polynomial function and since $f(1)$ is negative and $f(2)$ is positive, there is at least one real zero between $x = 1$ and $x = 2$.