given the values in the table on the right come from a continuous function, y has at least how many zeroes…

given the values in the table on the right come from a continuous function, y has at least how many zeroes? 0 1 2 3

given the values in the table on the right come from a continuous function, y has at least how many zeroes? 0 1 2 3

Answer

Explanation:

Step1: Use the Intermediate Value Theorem

The Intermediate Value Theorem states that if a function (y = 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 ((a,b)) such that (f(c)=k). When looking for zeros ((y = 0)), we check for sign - changes in the (y) - values.

Step2: Check sign - changes

  • Between (x=-3.2) ((y=-1.19)) and (x = - 2.4) ((y = 4.62)): The (y) - value changes from negative ((-1.19)) to positive ((4.62)). By the Intermediate Value Theorem, there is at least one zero in the interval ((-3.2,-2.4)).
  • Between (x=-0.8) ((y=-0.64)) and (x=-1.6) ((y = 3.78)): This is an error. We should check adjacent (x) - values. Between (x=-1.6) ((y = 3.78)) and (x=-0.8) ((y=-0.64)), the (y) - value changes from positive ((3.78)) to negative ((-0.64)). So, there is at least one zero in the interval ((-1.6,-0.8)).
  • Between (x = 2.4) ((y = 7.26)) and (x=1.6) ((y=-4.7)): The (y) - value changes from negative ((-4.7)) to positive ((7.26)). So, there is at least one zero in the interval ((1.6,2.4))

Answer:

2