find any critical numbers of the function. (enter your answers as a comma - separated list.)\ng(x)=x^{6}-3x^{…
find any critical numbers of the function. (enter your answers as a comma - separated list.)\ng(x)=x^{6}-3x^{4}\nx =
Answer
Explanation:
Step1: Differentiate the function
Using the power rule ((x^n)^\prime = nx^{n - 1}), if (g(x)=x^{6}-3x^{4}), then (g^\prime(x)=6x^{5}-12x^{3}).
Step2: Set the derivative equal to zero
Set (g^\prime(x) = 0), so (6x^{5}-12x^{3}=0). Factor out (6x^{3}): (6x^{3}(x^{2}-2)=0).
Step3: Solve for (x)
From (6x^{3}=0), we get (x = 0). From (x^{2}-2=0), we have (x^{2}=2), then (x=\pm\sqrt{2}).
Answer:
(0,-\sqrt{2},\sqrt{2})