Provide an appropriate response.Assume that f(x) and g(x) are two functions with the following properties: g(x) and f(x) are everywhere continuous, differentiable, and positive; f(x) is everywhere increasing and g(x) is everywhere decreasing. Which of the following functions are everywhere decreasing? Prove your assertions.i). h(x) = f(x) + g(x)ii). j(x) = f(x) ? g(x)iii). k(x) = iv). p(x) = g(x)v). r(x) = f(g(x)) = (f ? g)(x)

Question

What will be an ideal response?

Answer 1

i). h'(x) = f'(x) + g'(x) . The signs of the terms are (+) + (-), therefore h'(x) may be positive or negative. Without more information, we cannot determine where h(x) is increasing or decreasing.
ii). j'(x) = f'(x)g(x) + f(x)g'(x) . The signs of the terms are (+)(+) + (+)(-) , therefore j'(x) may be positive or negative. Without more information, we cannot determine where j(x) is increasing or decreasing.
iii). k'(x) = . The signs of the terms are  = (-). The quotient  is continuous and differentiable for all x, and k'(x) is everywhere negative. So, according to the first derivative test, k(x) is everywhere decreasing.
iv). p'(x) = (g(x))(^g(x)-1)(g'(x) ). The signs of the factors are (+)(+)(-) = (-). The function  ^g(x) is everywhere continuous and differentiable, and p'(x) is everywhere negative. So, according to the first derivative test, p'(x) is everywhere decreasing.
v). r'(x) = f'(g(x))g'(x). The signs of the factors are (+)(-) = (-). The function (f?g)(x) is everywhere continuous and differentiable, and r'(x) is everywhere negative. So, according to the first derivative test, r(x) is everywhere decreasing.