This assignment will count for 3 points on homework 1 and is due 9/05 at the beginning of class.
I expect clear thinking with COMPLETE sentences and proper grammar. If you wouldn’t
turn it in for an essay in a writing class, I won’t want it either.
1.
(3 points)
Let f be a continuous and strictly increasing function on [0, ∞) with f (0) = 0 and f (x) → ∞
as x → ∞. Use geometric reasoning to establish Young’s Inequality. For a > 0, b > 0,
Z
ab ≤
a
Z
f (x)dx +
0
b
f −1 (y)dy
0
[Hint(s): 1. Draw the picture(s). 2. The condition for equality is b = f (a).]