- Question 3 20 Marks Assume That G X Be A Function On 0 100 And Has A Derivative G X And A Primitive Function G X 1 (114.57 KiB) Viewed 27 times
and has a derivative g 0 (x) and a primitive function G(x) = R x 0
g(t)dt defined on the same domain. For all x ∈ (0, 100), it is
known that g(x) ≥ 0 and g 0 (x) ≤ 0. (1) In the 5th week, we
learned that g 0 (x) ≤ 0 for all x ∈ (0, 100) implies that g(x) is
decreasing. Show that this statement is true by using the
properties of definite integrals. In other words, for all a, b ∈
(0, 100), show that f(a) ≥ f(b) if a < b. (2) We also learned
that G00(x) = g 0 (x) ≤ 0 for all x ∈ (0, 100) implies that G(x) is
concave. Show that this statement is true by using the properties
of definite integrals and results from (1). In other words, for all
a, b ∈ (0, 100), show that G 1 2 (a + b) ≥ 1 2 (G(a) + G(b)). Hint:
for arbitrary continuous functions h(x) and f(x) on interval [a,
b], Z b a h(x)dx ≥ Z b a f(x)dx, if h(x) ≥ f(x) for all x ∈ [a,
b].
Question 3 (20 marks) Assume that g(x) be a function on (0,100) and has a derivative g'(x) and a primitive function G(x) = so g(t)dt defined on the same domain. For all x € (0, 100), it is known that g(x) > 0 and g'(x) < 0. (1) In the 5th week, we learned that g'(x) < 0 for all x € (0, 100) implies that g(x) is decreasing. Show that this statement is true by using the properties of definite integrals. In other words, for all a, b e (0, 100), show that f(a) f(b) if a < b. , = (2) We also learned that G" (x) = g'(x) < 0 for all x € (0,100) implies that G(x) is concave. Show that this statement is true by using the properties of definite integrals and results from (1). In other words, for all a, b € (0, 100), show that 62 (Ga) ((a+b)) G (a+b) ) = 5 (G(a) +G(b)). Hint: for arbitrary continuous functions h(x) and f(x) on interval [a, b], ["Hade 2 5° f(a)da, {) S* $(@adax, if h(x) = f(a) for all a € (a,b).