Mathematics
Mathematics
Problem
Construct a continuous function on the closed interval that attains its absolute maximum at even though the derivative at does not exist. Provide an explicit formula for such a function.Intuitive explanation
The key insight is that a function can have a "sharp corner" or "cusp" at its maximum point, where the derivative fails to exist because the left and right slopes don't match. Think of an absolute value function or a similar V-shaped (or inverted V-shaped) curve: it's perfectly continuous, but the sharp peak prevents differentiability there. You need a function that's smooth everywhere except at , where it reaches its highest value with a sharp point.
Intuitive explanation
The key insight is that a function can have a "sharp corner" or "cusp" at its maximum point, where the derivative fails to exist because the left and right slopes don't match. Think of an absolute value function or a similar V-shaped (or inverted V-shaped) curve: it's perfectly continuous, but the sharp peak prevents differentiability there. You need a function that's smooth everywhere except at , where it reaches its highest value with a sharp point.