The Intermediate Value Theorem only guarantees a zero if the function changes sign (from positive to negative or vice versa) over an interval. If both y-values are positive or both are negative, a zero is not guaranteed because the function might stay above or below the axis without crossing it.
Conditions: The function is continuous.; The y-values at the endpoints of the interval have the same sign.
The Intermediate Value Theorem only guarantees a zero if the function changes sign (from positive to negative or vice versa) over an interval. If both y-values are positive or both are negative, a zero is not guaranteed because the function might stay above or below the axis without crossing it.
Conditions: The function is continuous.; The y-values at the endpoints of the interval have the same sign.
If a continuous function takes a positive value at one x and a negative value at another x, there is at least one zero between those two x-values. This is because the graph must cross the x-axis (where y=0) to move from above the axis to below it without breaking.
Conditions: The function is continuous on the interval between the two x-values.; One endpoint value is positive and the other is negative.
If a continuous function takes a positive value at one x and a negative value at another x, there is at least one zero between those two x-values. This is because the graph must cross the x-axis (where y=0) to move from above the axis to below it without breaking.
Conditions: The function is continuous on the interval between the two x-values.; One endpoint value is positive and the other is negative.