🟦 Máximos y Mínimos de una Función (Criterio de la Primer Derivada) | Video 2

Summary

In this video, the steps for finding the maximum and minimum points of a polynomial function are thoroughly explained. The process includes calculating the derivative, setting it equal to zero to find critical points, and analyzing the sign changes of the derivative around these points. The function’s maximum occurs at x = 1, and its minimum occurs at x = 3. Finally, the coordinates of the maximum and minimum points are calculated, with the maximum at (1, 4) and the minimum at (3, 0). The video offers a clear, step-by-step guide for solving this type of problem in calculus.

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Q & A

What is the first step in the exercise?

The first step is to calculate the derivative of the given function, f(x). This is done by finding the derivative of each term in the polynomial.

How is the derivative of f(x) calculated?

The derivative of f(x) is calculated by applying the power rule. For example, the derivative of x^3 is 3x^2, the derivative of -6x^2 is -12x, and the derivative of 9x is 9.

What is the second step in solving this exercise?

The second step is to set the first derivative equal to zero in order to find the critical points of the function.

How is the equation simplified before solving for the critical points?

The equation is simplified by dividing both sides by 3 to make the terms easier to work with. This leads to the equation x^2 - 4x + 3 = 0.

What method is used to solve the quadratic equation?

The quadratic equation is solved by factoring. The equation x^2 - 4x + 3 is factored into (x - 3)(x - 1) = 0.

What are the critical points found in the exercise?

The critical points are x = 3 and x = 1, which are the solutions to the factored equation.

What is the purpose of evaluating the first derivative at points around the critical points?

The purpose of evaluating the first derivative at points around the critical points is to determine whether the critical points are maxima or minima.

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

By using the first derivative test: if the derivative changes from positive to negative at a point, it’s a maximum; if it changes from negative to positive, it’s a minimum.

What are the results of evaluating the first derivative at x = 0 and x = 2?

When evaluating the first derivative at x = 0, the result is 9, and at x = 2, the result is -3. The sign changes from positive to negative, indicating a maximum at x = 1.

What are the coordinates of the maximum and minimum points of the function?

The function has a maximum at (1, 4) and a minimum at (3, 0), as determined by evaluating the function at the critical points.