MVT & Concavity Review

Front

If f'(x) < 0 over an interval, then f(x) is ____________ over that interval.

Back

decreasing

Front

What does a hole in the graph of a function indicate about continuity at that point?

Back

It indicates that the function is not defined at that point, leading to a discontinuity.

Front

If (a,b) is a local minimum, what can be said about f'(a)?

Back

f'(a) = 0

Front

When f'(x) changes from positive to negative, what does this indicate?

Back

There is a maximum at that point.

Front

If g''(3) = -8, what does this tell us about g(x) at x=3?

Back

g(x) is concave down at x=3.

Front

What does it mean if f''(x) > 0 on an interval?

Back

The function f(x) is concave up on that interval.

Front

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

Back

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

Front

What is a critical point?

Back

A critical point occurs where f'(x) = 0 or f'(x) is undefined.

Front

What does f''(x) = 0 indicate about the function?

Back

It may indicate a point of inflection, but further analysis is needed.

Front

What is the significance of a local maximum?

Back

A local maximum is a point where the function value is higher than all nearby points.