increasing decreasing functions and first derivative test

Front

What is a local maximum in a function?

Back

A local maximum is a point in the domain of a function where the function value is greater than the values of the function at nearby points.

Front

What does f''(a) indicate if (a,b) is a local maximum?

Back

f''(a) is negative.

Front

What is a critical point?

Back

A critical point is a point in the domain of a function where the derivative is zero or undefined.

Front

True or false: all critical points are extrema.

Back

False.

Front

What is the First Derivative Test?

Back

The First Derivative Test is a method used to determine whether a critical point is a local maximum, local minimum, or neither by analyzing the sign of the first derivative before and after the critical point.

Front

What does it mean if f'(x) changes from positive to negative at x = c?

Back

It indicates that there is a local maximum at x = c.

Front

What does it mean if f'(x) changes from negative to positive at x = c?

Back

It indicates that there is a local minimum at x = c.

Front

What is concavity in relation to a function?

Back

Concavity refers to the direction in which a function curves: concave up (curves upwards) or concave down (curves downwards).

Front

How do you determine intervals of concavity?

Back

Intervals of concavity are determined by analyzing the second derivative of the function: if f''(x) > 0, the function is concave up; if f''(x) < 0, the function is concave down.

Front

What is the significance of the second derivative test?

Back

The second derivative test helps determine the nature of critical points: if f''(c) > 0, then f has a local minimum at c; if f''(c) < 0, then f has a local maximum at c.