04. Maximum, minimum, growth and decrease: criterion of the first derivative

Summary

In this video, the process of determining the maximum and minimum points, as well as growth and decay intervals for the function f(x) = x^4 - 2x^2 + 3, is thoroughly explained. The presenter walks through calculating the first derivative, finding critical points, and analyzing the function's behavior using a number line. Key steps include identifying intervals of positivity and negativity in the derivative, which reveals where the function is increasing or decreasing. The video also covers how to identify the points of maximum and minimum using the first derivative test, and ends with a graphical representation of the function's behavior.

Q & A

What is the first step in determining maximums, minimums, growth, and decay of the function?

The first step is to calculate the derivative of the function, also known as f'(x).

How do you calculate the critical points for this function?

To calculate the critical points, you set the derivative equal to 0 and solve the equation. In this case, the derivative of the function is 4x³ - 4x, which is set equal to 0.

What is the derivative of the function f(x) = x⁴ - 2x² + 3?

The derivative is f'(x) = 4x³ - 4x.

How do you find the values of x where the derivative equals 0?

To find the values of x where the derivative equals 0, factor the derivative expression. In this case, 4x³ - 4x factors as 4x(x² - 1), and setting each factor equal to 0 gives x = 0, x = 1, and x = -1.

What is the significance of the points where the derivative equals 0?

The points where the derivative equals 0 are the critical points, which could correspond to maxima, minima, or points of inflection.

What method is used to determine whether a point is a maximum, minimum, or neither?

The first derivative test is used, which involves checking how the sign of the derivative changes around the critical points. If the derivative changes from positive to negative, it's a maximum. If it changes from negative to positive, it's a minimum.

What is the behavior of the function in the intervals determined by the critical points?

In the interval (-∞, -1), the function is decreasing. In the interval (-1, 0), the function is increasing. In the interval (0, 1), the function is decreasing again. In the interval (1, ∞), the function is increasing.

How do you determine if the function is increasing or decreasing?

You substitute values from each interval into the derivative and check if the result is positive (increasing) or negative (decreasing). For example, substituting -2 into the derivative gives a negative result, indicating the function is decreasing in that interval.

What is the first critical point found in this example and what type of point is it?

The first critical point is x = -1, and it is a minimum because the derivative changes from negative to positive around this point.

How do you find the coordinates of the critical points to graph the function?

To find the coordinates of the critical points, substitute the x-values of the critical points (x = -1, x = 0, and x = 1) into the original function f(x). For example, f(-1) = 2, so the coordinates of the point are (-1, 2).