determine the critical values for the function: f(x)=x^4 - 50x^2 + 100 x = ?, , enter the smallest value

Explanation:

Step1: Find the derivative

The derivative of $f(x)=x^{4}-50x^{2}+100$ using the power - rule $(x^n)'=nx^{n - 1}$ is $f'(x)=4x^{3}-100x$.

Step2: Set the derivative equal to zero

Set $f'(x)=0$, so $4x^{3}-100x = 0$. Factor out $4x$: $4x(x^{2}-25)=0$.

Step3: Solve the factored equation

We have two cases from $4x(x^{2}-25)=0$. Case 1: $4x=0$, which gives $x = 0$. Case 2: $x^{2}-25=0$, which can be written as $(x + 5)(x - 5)=0$. So $x=-5$ or $x = 5$.

Answer:

$-5$