Question

show that the function ( f(x)=x^{4}+5x + 3 ) has exactly one zero in the interval (-1,0).
which theorem can be used to determine whether a function ( f(x) ) has any zeros in a given interval?
a. intermediate value theorem
b. extreme value theorem
c. rolles theorem
d. mean value theorem

Explanation:

The Intermediate Value Theorem states that if a function \( f(x) \) is continuous on a closed interval \([a,b]\), and \( k \) is a number between \( f(a) \) and \( f(b) \), then there exists at least one number \( c\) in the interval \((a,b)\) such that \( f(c)=k \). When checking for zeros, we set \( k = 0 \). The Extreme Value Theorem is about the existence of maximum and minimum values on a closed interval. Rolle's Theorem requires \( f(a)=f(b) \) and gives a condition for a derivative to be zero. The Mean Value Theorem relates the average rate of change to the instantaneous rate of change. Since we are checking for the existence of a zero (i.e., \( f(x)=0 \)) in an interval, the Intermediate Value Theorem is the relevant one.

Answer:

A. Intermediate value theorem