Questions: Find the critical points for: f(x)=x^3-3x-9x+7

Transcript text: Find the critical points for: \[ f(x)=x^{3}-3 x-9 x+7 \]

Solution

Solution Steps

To find the critical points of the function \( f(x) = x^3 - 3x - 9x + 7 \), we need to follow these steps:

Compute the first derivative of the function, \( f'(x) \).
Set the first derivative equal to zero and solve for \( x \) to find the critical points.
Verify the critical points by checking the second derivative or using the first derivative test.

Step 1: Define the Function

We start with the function given by \[ f(x) = x^3 - 3x - 9x + 7 = x^3 - 12x + 7. \]

Step 2: Compute the First Derivative

Next, we compute the first derivative of the function: \[ f'(x) = \frac{d}{dx}(x^3 - 12x + 7) = 3x^2 - 12. \]

Step 3: Find Critical Points

To find the critical points, we set the first derivative equal to zero: \[ 3x^2 - 12 = 0. \] Solving for \( x \), we get: \[ 3x^2 = 12 \implies x^2 = 4 \implies x = \pm 2. \] Thus, the critical points are \( x = -2 \) and \( x = 2 \).

Final Answer

The critical points are \[ \boxed{x = -2} \quad \text{and} \quad \boxed{x = 2}. \]

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