show that the function ( f(x)=x^{4}+5x + 3 ) has exactly one zero in the interval (-1,0).\nwhich theorem can…
show that the function ( f(x)=x^{4}+5x + 3 ) has exactly one zero in the interval (-1,0).\nwhich theorem can be used to determine whether a function ( f(x) ) has any zeros in a given interval?\na. intermediate value theorem\nb. extreme value theorem\nc. rolles theorem\nd. mean value theorem
Answer
Brief Explanations:
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