Let f(x)=4x2−2x4. Find the open intervals on which f is increasing (decreasing). Then determine the x-coordinates of all relative maxima (minima).

Answer 1

y' = 8x - 8x^3 = 8x( 1 - x^2) = 8x( 1+x)(1-x)

extrema occur at x=0, x=-1, and x=1

interval (-infinity, -1) ---> function is increasing because the derivative is positive

interval (-1, 0) ---> function is decreasing because the derivative is negative

interval (0,1) --> function is increasing because the derivative is positive

interval (1,infinity) ---> function is decreasing because the derivative is negative