Question

Let f ( x ) = x 4 − 2 x 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

f(x)=x⁴-2x², f'(x)=4x³-4x=0=4x(x-1)(x+1), so c=-1,0,1 all in (-2,2).

f(-2)=f(2)=16-8=8, so Rolle’s Theorem says there must a value c between -2 and 2 where the gradient f' must be zero (flat). Also f(0)=0 so between -2 and 0, 0 and 2 there must also be zero gradients f(-√2)=f(√2)=f(0)=0.

The three intercepts for the gradient concur with Rolle’s Theorem since -1 lies between -√2 and 0, 0 lies between -2 and 2, and 1 lies between 0 and √2.