Question

Let f:R→R be defined as f(x) = |x| – |x³ – 1| If f has local maximum at two points, find the distance between them.

Answer 1

We have f(x) = |x| – |x³ – 1|

Now, for max or min, f¢(x) = 0 gives

x = –√(2/3), 0, √(2/3)

Thus, the function provide us the point of local max are x = – √(2/3), √(2/3)

when x = – √(2/3), then y = 1/3√2/3 –1

when x = √2/3 , then y = 1/3 √(2/3)  + 1

Thus, the points of local maximum are

P( –√(2/3), 1/3√(2/3)  – 1) and Q(√(2/3), 1/3√(2/3)  + 1)

Hence, the distance between P and Q is