increasing decreasing functions and first derivative test

Front

What is an increasing function?

Back

A function f(x) 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

What is a decreasing function?

Back

A function f(x) 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 a critical point?

Back

A critical point of a function f(x) is a point x where f'(x) = 0 or f'(x) is undefined.

Front

What does the first derivative test determine?

Back

The first derivative test is used to determine whether a critical point is a local maximum, local minimum, or neither.

Front

What is the relationship between f'(x) and increasing/decreasing functions?

Back

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

Front

What is a local maximum?

Back

A local maximum is a point (a, b) on the graph of f(x) where f(a) is greater than f(x) for all x in some interval around a.

Front

What is a local minimum?

Back

A local minimum is a point (a, b) on the graph of f(x) where f(a) is less than f(x) for all x in some interval around a.

Front

What does f''(x) indicate about concavity?

Back

If f''(x) > 0, the function is concave up; if f''(x) < 0, the function is concave down.

Front

What is the significance of f''(a) at a local maximum?

Back

If (a, b) is a local maximum, then f''(a) is negative.

Front

What is the significance of f''(a) at a local minimum?

Back

If (a, b) is a local minimum, then f''(a) is positive.