Derivative Behaviors (f, f', f")

Front

What is a critical point in the context of derivatives?

Back

A critical point occurs where the first derivative of a function is zero or undefined, indicating potential local maxima, minima, or inflection points.

Front

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

Back

It indicates a potential inflection point where the concavity of the function may change.

Front

How does the first derivative f'(x) relate to the behavior of the function f(x)?

Back

The first derivative indicates the slope of the function; if f'(x) > 0, the function is increasing; if f'(x) < 0, the function is decreasing.

Front

What does the second derivative f''(x) tell us about the concavity of a function?

Back

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

Front

What is an inflection point?

Back

An inflection point is a point on the curve where the concavity changes, typically where f''(x) = 0 and there is a sign change.

Front

If f'(-3) = 5, what can we conclude about the function f(x) at x = -3?

Back

The function f(x) is increasing at x = -3.

Front

What is the relationship between the first and second derivatives in determining concavity?

Back

The second derivative (f'') is used to determine the concavity of the function based on the sign of f''.

Front

What does it mean if f''(x) changes sign at a point?

Back

It indicates that there is an inflection point at that x value.

Front

What is the significance of a local maximum in relation to the first derivative?

Back

At a local maximum, the first derivative changes from positive to negative.

Front

What is the significance of a local minimum in relation to the first derivative?

Back

At a local minimum, the first derivative changes from negative to positive.