Questions: Find the critical numbers of the function. f(x)=x^3+3 x^2-45 x x=

Find the critical numbers of the function. f(x)=x^3+3 x^2-45 x x=
Transcript text: Find the critical numbers of the function. \[ \begin{array}{r} f(x)=x^{3}+3 x^{2}-45 x \\ x=\square \end{array} \]
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Solution

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Solution Steps

To find the critical numbers of the function \( f(x) = x^3 + 3x^2 - 45x \), we need to follow these steps:

  1. Compute the first derivative of the function, \( f'(x) \).
  2. Set the first derivative equal to zero and solve for \( x \).
  3. The solutions to this equation are the critical numbers.
Step 1: Find the First Derivative

We start with the function \( f(x) = x^3 + 3x^2 - 45x \). The first derivative is calculated as follows: \[ f'(x) = 3x^2 + 6x - 45 \]

Step 2: Set the First Derivative to Zero

To find the critical numbers, we set the first derivative equal to zero: \[ 3x^2 + 6x - 45 = 0 \]

Step 3: Solve for Critical Numbers

We can simplify the equation by dividing all terms by 3: \[ x^2 + 2x - 15 = 0 \] Next, we factor the quadratic: \[ (x - 3)(x + 5) = 0 \] Setting each factor to zero gives us the critical numbers: \[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \] \[ x + 5 = 0 \quad \Rightarrow \quad x = -5 \]

Final Answer

The critical numbers of the function are \\(\boxed{-5, 3}\\).

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