Question

Let \\(f(x)=x^{4}-2x^{2}\\). Find the all \\(c\\) (where \\(c\\) is the interception on the x-axis ) in the interval (-2, 2) such that \\(f\'(x)=0\\). ( Hint use Rolle\'s theorem )

Answer 1

Function $f(x)=x^4-2x^2$  is continuous on the closed interval [-2, 2] and differentiable on the open interval (-2, 2).

$f(-2)=f(2)=8$

According to Rolle's theorem, there should be at least one point c in the open interval (a, b) for which f'(c)=0.

Let's find f'(x)

$f'(x)=4x^3-4x$

Therefore, we get the equation

$4x^3-4x=0$

$x(x^2-1)=0$

$x=-1;x=0;x=1$

All of these roots belong to the open interval (-2, 2), hence the answer is -1; 1; 0.