Questions: Find the critical point(s) within the domain of the following function. Answer exactly. y=25 x^3-12 x

Find the critical point(s) within the domain of the following function. Answer exactly. y=25 x^3-12 x
Transcript text: Find the critical point(s) within the domain of the following function. Answer exactly. \[ \begin{array}{l} y=25 x^{3}-12 x \\ \end{array} \]
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Solution

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Solution Steps

To find the critical points of the function \( y = 25x^3 - 12x \), we need to follow these steps:

  1. Compute the derivative of the function with respect to \( x \).
  2. Set the derivative equal to zero to find the critical points.
  3. Solve the resulting equation for \( x \).
Step 1: Compute the Derivative

We start with the function \( y = 25x^3 - 12x \). The derivative of this function is calculated as follows: \[ \frac{dy}{dx} = 75x^2 - 12 \]

Step 2: Set the Derivative to Zero

To find the critical points, we set the derivative equal to zero: \[ 75x^2 - 12 = 0 \]

Step 3: Solve for \( x \)

Rearranging the equation gives: \[ 75x^2 = 12 \] \[ x^2 = \frac{12}{75} = \frac{4}{25} \] Taking the square root of both sides, we find: \[ x = \pm \frac{2}{5} \]

Final Answer

The critical points of the function are: \[ \boxed{x = -\frac{2}{5}, \frac{2}{5}} \]

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