use the intermediate value theorem to confirm that the given polynomial has at least one zero within the…

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

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

Answer

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 $x = 2$.