determine the critical values for the function: f(x)=x^4 - 50x^2 + 100 x=-5,0,? enter the largest value.

Explanation:

Step1: Find the derivative

$f'(x)=4x^{3}-100x$

Step2: Set the derivative equal to zero

$4x^{3}-100x = 0$ Factor out $4x$: $4x(x^{2}-25)=0$ Using the difference - of - squares formula $a^{2}-b^{2}=(a + b)(a - b)$ where $a=x$ and $b = 5$, we have $4x(x + 5)(x - 5)=0$

Step3: Solve for x

$4x=0$ gives $x = 0$; $x+5=0$ gives $x=-5$; $x - 5=0$ gives $x = 5$

Answer:

5