First and Second Derivative Test

Front

What is a critical point in calculus?

Back

A critical point of a function is where its derivative is zero or undefined. It is a point where the function may have a local maximum, local minimum, or a point of inflection.

Front

How do you find critical points of a function?

Back

To find critical points, take the derivative of the function, set it equal to zero, and solve for x. Also, check where the derivative 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 by analyzing the sign of the derivative before and after the critical point.

Front

What does the Second Derivative Test determine?

Back

The Second Derivative Test determines the concavity of the function at a critical point. If f''(x) > 0, the point is a local minimum; if f''(x) < 0, it is a local maximum.

Front

What is a local maximum?

Back

A local maximum is a point where the function value is higher than all nearby points. It occurs at a critical point where the derivative changes from positive to negative.

Front

What is a local minimum?

Back

A local minimum is a point where the function value is lower than all nearby points. It occurs at a critical point where the derivative changes from negative to positive.

Front

What is concavity in calculus?

Back

Concavity refers to the direction of the curvature of a function. A function is concave up if its second derivative is positive and concave down if its second derivative is negative.

Front

How do you determine intervals of increase and decrease?

Back

To determine intervals of increase and decrease, analyze the sign of the first derivative. If f'(x) > 0, the function is increasing; if f'(x) < 0, it is decreasing.

Front

What is an inflection point?

Back

An inflection point is a point on the curve where the concavity changes. It occurs where the second derivative is zero or undefined.

Front

What is the significance of a sign chart?

Back

A sign chart is a visual tool used to determine the sign of a function or its derivatives over intervals, helping to identify critical points, local extrema, and concavity.