let f(x)=-x^4 + 6x^2 - 3. find all critical points of f. as your answer, please input the sum of all…
let f(x)=-x^4 + 6x^2 - 3. find all critical points of f. as your answer, please input the sum of all critical points.
Answer
Explanation:
Step1: Find the derivative
The derivative of $f(x)=-x^{4}+6x^{2}-3$ using the power - rule $(x^n)' = nx^{n - 1}$ is $f'(x)=-4x^{3}+12x$.
Step2: Set the derivative equal to zero
Set $f'(x)=0$, so $-4x^{3}+12x = 0$. Factor out $-4x$ to get $-4x(x^{2}-3)=0$.
Step3: Solve for x
From $-4x(x^{2}-3)=0$, we have two cases. Case 1: $-4x = 0$, which gives $x = 0$. Case 2: $x^{2}-3=0$, which gives $x=\pm\sqrt{3}$.
Step4: Calculate the sum of critical points
The critical points are $x = 0,x=\sqrt{3},x = -\sqrt{3}$. The sum is $0+\sqrt{3}+(-\sqrt{3})=0$.
Answer:
0