First Derivative Test

Front

What is the First Derivative Test?

Back

The First Derivative Test is a method used to determine the local maxima and minima of a function by analyzing the sign of its first derivative.

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, consider points where the derivative does not exist.

Front

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

Back

If the first derivative is positive at a point, the function is increasing at that point.

Front

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

Back

If the first derivative is negative at a point, the function is decreasing at that point.

Front

What is a relative maximum?

Back

A relative maximum is a point where the function value is higher than the values of the function at nearby points.

Front

What is a relative minimum?

Back

A relative minimum is a point where the function value is lower than the values of the function at nearby points.

Front

How do you determine if a critical point is a maximum or minimum?

Back

Use the First Derivative Test: if the derivative changes from positive to negative, it's a maximum; if it changes from negative to positive, it's a minimum.

Front

What is the significance of the second derivative in relation to the First Derivative Test?

Back

The second derivative can provide information about the concavity of the function, helping to confirm whether a critical point is a maximum or minimum.

Front

Find the critical points of f(x) = 2x^4 - 4x^2 + 1.

Back

The critical points are x = -1, 0, 1.

Front

If f'(x) < 0 on an interval, what can be said about f(x)?

Back

If f'(x) < 0 on an interval, then f(x) is decreasing on that interval.