3. (14 points) Suppose f : [0,00) + R is convex, continuously differentiable, and f(0) = 0. Show that for any -10, f(x) 1*2*exp( 11%de 5 exp(a + 1) de s exp(a + 1) [ ***exp(-:'(2) dr. (Hint: Consider change of variable r = py in the left hand side for p slightly bigger than 1, and compare f(py) to f(y) + (p-1)yf'(y). Apply Hlder, and then let p \ 1.) 3. (14 points) Suppose f : [0,00) + R is convex, continuously differentiable, and f(0) = 0. Show that for any -10, f(x) 1*2*exp( 11%de 5 exp(a + 1) de s exp(a + 1) [ ***exp(-:'(2) dr. (Hint: Consider change of variable r = py in the left hand side for p slightly bigger than 1, and compare f(py) to f(y) + (p-1)yf'(y). Apply Hlder, and then let p \ 1.)