use the intermediate value theorem to determine whether the given polynomial has at least one zero within…

use the intermediate value theorem to determine whether the given polynomial has at least one zero within the given interval.\na) ( f(x)=-x^{4}+4 ) between ( x = 1 ) and 3.\no no, there is no zero between ( x = 1 ) and 3.\no unable to determine.\no yes, there is at least one zero between ( x = 1 ) and 3.

use the intermediate value theorem to determine whether the given polynomial has at least one zero within the given interval.\na) ( f(x)=-x^{4}+4 ) between ( x = 1 ) and 3.\no no, there is no zero between ( x = 1 ) and 3.\no unable to determine.\no yes, there is at least one zero between ( x = 1 ) and 3.

Answer

Explanation:

Step1: Calcular ( f(1) )

Sustituir ( x = 1 ) en ( f(x)=-x^{4}+4 ): ( f(1)=-(1)^{4}+4=-1 + 4=3 )

Step2: Calcular ( f(3) )

Sustituir ( x = 3 ) en ( f(x)=-x^{4}+4 ): ( f(3)=-(3)^{4}+4=-81 + 4=-77 )

Step3: Aplicar el Teorema del Valor Intermedio

El Teorema del Valor Intermedio establece que si una función ( y = f(x) ) es continua en un intervalo cerrado ([a,b]), y ( k ) es un número entre ( f(a) ) y ( f(b) ), entonces existe al menos un número ( c) en el intervalo ((a,b)) tal que ( f(c)=k ). En este caso, la función ( f(x)=-x^{4}+4 ) es un polinomio, por lo que es continua en ( \mathbb{R} ), en particular en ([1,3]). Como ( f(1)=3>0 ) y ( f(3)=-77<0 ), y ( 0 ) está entre ( f(1) ) y ( f(3) ), entonces existe al menos un ( c\in(1,3) ) tal que ( f(c) = 0 ).

Answer:

Yes, there is at least one zero between ( x = 1 ) and ( 3 ).