A function f(x) = x^3 + 6x^2 -3x + 4 has a critical point at

To find the critical points of the function \( f(x) = x^3 + 6x^2 - 3x + 4 \), we need to follow these steps:

1. **Find the derivative of the function**: 
   \[
   f'(x) = \frac{d}{dx}(x^3 + 6x^2 - 3x + 4)
   \]

2. **Set the derivative equal to zero and solve for \( x \)**:
   \[
   f'(x) = 3x^2 + 12x - 3
   \]
   To find the critical points, set the derivative equal to zero:
   \[
   3x^2 + 12x - 3 = 0
   \]

3. **Solve the quadratic equation**:
   Divide the entire equation by 3:
   \[
   x^2 + 4x - 1 = 0
   \]
   To solve this quadratic equation, we use the quadratic formula:
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   where \( a = 1 \), \( b = 4 \), and \( c = -1 \).

4. **Substitute the values into the quadratic formula**:
   \[
   x = \frac{-4 \pm \sqrt{(4)^2 - 4(1)(-1)}}{2(1)}
   \]
   \[
   x = \frac{-4 \pm \sqrt{16 + 4}}{2}
   \]
   \[
   x = \frac{-4 \pm \sqrt{20}}{2}
   \]
   \[
   x = \frac{-4 \pm 2\sqrt{5}}{2}
   \]
   \[
   x = -2 \pm \sqrt{5}
   \]

So, the critical points occur at:
\[
x = -2 + \sqrt{5} \quad \text{and} \quad x = -2 - \sqrt{5}
\]

These are the critical points of the function \( f(x) \).

Would you like more details on any of these steps, or do you have any questions?

Here are some related questions:

1. How do you classify the nature of critical points (local maxima, minima, or saddle points)?
2. What is the importance of the second derivative in determining the concavity of a function?
3. How do you solve higher-degree polynomials to find critical points?
4. What are inflection points, and how do you find them?
5. How does the shape of the graph of a cubic function typically behave around critical points?

**Tip:** Always check the second derivative at critical points to determine whether they correspond to local maxima, minima, or points of inflection.

Discover how to find critical points of the function f(x) = x^3 + 6x^2 - 3x + 4 with a detailed step-by-step solution. This problem covers key calculus concepts, including derivatives and critical points. Suitable for advanced high school students.

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