Question

4. Suppose f(x) is a continuous function on [0,∞) and f(x) > 0 for all x ∈ [0,∞). Let

􏰀 x tf(t)dt 0

g(x) = 􏰀 x . Show that 0 f(t)dt

(1) lim g(x) = 0, and x→0+

1

(2) g(x) is increasing on (0, ∞).

4. Suppose f(x) is a continuous function on [0,00) and f(x) > 0 for all x [0,00). Let g(z) = f뜻블 show that (1) lim g(x)=0, and (t)dt (2) g(x) is increasing on (0, oo)

Show transcribed image text 4. Suppose f(x) is a continuous function on [0,00) and f(x) > 0 for all x [0,00). Let g(z) = f뜻블 show that (1) lim g(x)=0, and (t)dt (2) g(x) is increasing on (0, oo)