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

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 \( x \).

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

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

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