Positive/Negative and Increasing/Decreasing Functions

Front

What is a decreasing function?

Back

A function is decreasing on an interval if, for any two points x1 and x2 in that interval, if x1 < x2 then f(x1) > f(x2).

Front

What is an increasing function?

Back

A function is increasing on an interval if, for any two points x1 and x2 in that interval, if x1 < x2 then f(x1) < f(x2).

Front

How do you identify where a function is positive?

Back

A function is positive where its output values (f(x)) are greater than zero.

Front

How do you identify where a function is negative?

Back

A function is negative where its output values (f(x)) are less than zero.

Front

What does it mean for a function to be positive on an interval?

Back

A function is positive on an interval if all output values (f(x)) are greater than zero for all x in that interval.

Front

What does it mean for a function to be negative on an interval?

Back

A function is negative on an interval if all output values (f(x)) are less than zero for all x in that interval.

Front

What is the significance of critical points in determining increasing/decreasing behavior?

Back

Critical points are where the derivative of the function is zero or undefined, and they help identify intervals where the function changes from increasing to decreasing or vice versa.

Front

How can you determine where a function is increasing or decreasing using its derivative?

Back

If the derivative f'(x) > 0 on an interval, the function is increasing; if f'(x) < 0, the function is decreasing.

Front

What is the relationship between the first derivative test and increasing/decreasing functions?

Back

The first derivative test uses the sign of the derivative to determine whether a function is increasing or decreasing at critical points.

Front

What is the interval notation for a function that is positive for x < -5 and 0 < x < 6?

Back

(-∞, -5) ∪ (0, 6) is the interval notation for where the function is positive.