Questions: Find the critical numbers of y=3+4x-5x^3.

Transcript text: Find the critical numbers of $y=3+4 x-5 x^{3}$.

Solution

Solution Steps

To find the critical numbers of the function $ y = 3 + 4x - 5x^3 $, we need to follow these steps:

Compute the first derivative of the function.
Set the first derivative equal to zero and solve for $ x $.
The solutions to this equation are the critical numbers.

Step 1: Find the First Derivative

We start with the function $ y = 3 + 4x - 5x^3 $. The first derivative is calculated as follows: \[ \frac{dy}{dx} = 4 - 15x^2 \]

Step 2: Set the Derivative to Zero

To find the critical numbers, we set the first derivative equal to zero: \[ 4 - 15x^2 = 0 \]

Step 3: Solve for $ x $

Rearranging the equation gives: \[ 15x^2 = 4 \] \[ x^2 = \frac{4}{15} \] Taking the square root of both sides, we find: \[ x = \pm \sqrt{\frac{4}{15}} = \pm \frac{2\sqrt{15}}{15} \]

Final Answer

The critical numbers are: \[ \boxed{x = -\frac{2\sqrt{15}}{15}, \frac{2\sqrt{15}}{15}} \]

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