5.3 The Second Derivative Test

Front

What is the second derivative of a polynomial function?

Back

The second derivative of a polynomial function provides information about the concavity of the function and can be used to determine points of inflection.

Front

How do you find the second derivative of a function?

Back

To find the second derivative, first find the first derivative of the function, then differentiate that result.

Front

What does it mean if the second derivative is positive at a point?

Back

If the second derivative is positive at a point, the function is concave up at that point.

Front

What does it mean if the second derivative is negative at a point?

Back

If the second derivative is negative at a point, the function is concave down at that point.

Front

What are critical numbers?

Back

Critical numbers are values of x where the first derivative is zero or undefined, indicating potential local maxima or minima.

Front

What is a point of inflection?

Back

A point of inflection is where the concavity of the function changes, which occurs where the second derivative is zero or undefined.

Front

How can you determine if a function has a relative minimum using the second derivative test?

Back

If the second derivative is positive at a critical number, the function has a relative minimum at that point.

Front

How can you determine if a function has a relative maximum using the second derivative test?

Back

If the second derivative is negative at a critical number, the function has a relative maximum at that point.

Front

What is the significance of g''(x) < 0?

Back

If g''(x) < 0, the function g(x) is concave down at that point.

Front

What is the significance of g''(x) > 0?

Back

If g''(x) > 0, the function g(x) is concave up at that point.