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Actividad 4. Máximos y mínimos. Criterio de la primera derivada

(Valor 5%) Determina los puntos máximos y mínimos para la función   utiliza el criterio de la primera derivada.


To find the maximum and minimum points of the function \(f(x) = x^3 - 3x\) using the first derivative test, we need to follow these steps:

1. **Find the first derivative** of the function.
2. **Determine the critical points** by setting the first derivative equal to zero.
3. **Analyze the sign of the first derivative** around the critical points to determine whether each point is a local maximum, minimum, or neither.

Let's go through these steps:

### 1. Find the First Derivative

Given the function:
\[
f(x) = x^3 - 3x
\]

The first derivative \(f'(x)\) is:
\[
f'(x) = \frac{d}{dx}(x^3) - \frac{d}{dx}(3x) = 3x^2 - 3
\]

### 2. Find the Critical Points

Set \(f'(x) = 0\) to find the critical points:
\[
3x^2 - 3 = 0
\]

Simplify and solve for \(x\):
\[
x^2 = 1 \implies x = \pm 1
\]

So, the critical points are \(x = 1\) and \(x = -1\).

### 3. Analyze the Sign of the First Derivative

To determine whether these critical points are maxima, minima, or neither, we analyze the sign of \(f'(x)\) on intervals around the critical points.

- For \(x < -1\): Choose \(x = -2\)
  \[
  f'(-2) = 3(-2)^2 - 3 = 12 - 3 = 9 \quad (\text{positive})
  \]
  
- For \(-1 < x < 1\): Choose \(x = 0\)
  \[
  f'(0) = 3(0)^2 - 3 = -3 \quad (\text{negative})
  \]

- For \(x > 1\): Choose \(x = 2\)
  \[
  f'(2) = 3(2)^2 - 3 = 12 - 3 = 9 \quad (\text{positive})
  \]

### Conclusion

- At \(x = -1\): \(f'(x)\) changes from positive to negative, indicating a **local maximum**.
- At \(x = 1\): \(f'(x)\) changes from negative to positive, indicating a **local minimum**.

### Maximum and Minimum Points

To find the actual points, we substitute \(x = -1\) and \(x = 1\) back into the original function \(f(x)\):

- \(f(-1) = (-1)^3 - 3(-1) = -1 + 3 = 2\)
  
- \(f(1) = (1)^3 - 3(1) = 1 - 3 = -2\)

Thus, the function \(f(x) = x^3 - 3x\) has:

- A local **maximum** at \((-1, 2)\)
- A local **minimum** at \((1, -2)\)

Would you like more details or have any questions about the solution?

### Further Related Questions

1. How do you find inflection points of a function using the second derivative?
2. What does the second derivative test tell us about a function's graph?
3. How can we use derivatives to sketch the graph of a function?
4. What are the global maximum and minimum of the function \(f(x) = x^3 - 3x\) on a closed interval?
5. How does concavity relate to the behavior of a function's first derivative?

### Tip

Remember, the first derivative test is a great way to determine local maxima and minima, but for more complex functions, consider using the second derivative test for additional insight into the function's concavity and inflection points.

Learn how to determine maximum and minimum points of a function using the first derivative test. Solve the problem for the function f(x) = x^3 - 3x. Includes step-by-step analysis and critical point identification.

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