Problem

The function given by $f(x)=\left(x^{2}-2\right)^{3}$ has exactly one positive critical number. What is that positive critical number? Note: We are asking for the $x$-coordinate. Enter your answer as a decimal. Round to three decimal places (as needed).

Solution

Step 1 :The function is given by \( f(x) = (x^2 - 2)^3 \)

Step 2 :To find the critical number, we need to take the derivative of the function and set it equal to zero.

Step 3 :The derivative of the function is \( f'(x) = 6x(x^2 - 2)^2 \)

Step 4 :We set the derivative equal to zero to find the critical points: \( 6x(x^2 - 2)^2 = 0 \)

Step 5 :The critical points are \( x = 0 \), \( x = -\sqrt{2} \), and \( x = \sqrt{2} \)

Step 6 :We are looking for the positive critical number, which is \( x = \sqrt{2} \)

Step 7 :The positive critical number rounded to three decimal places is \( \boxed{1.414} \)

From Solvely APP
Source: https://solvelyapp.com/problems/XnTXUxcbmD/

Get free Solvely APP to solve your own problems!

solvely Solvely
Download